Is limited liability or double taxation fair?

WEBVTT
Kind: captions
Language: en

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Voiceover: After watching the last video

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on corporations and limited liability,

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you might be asking yourself was that fair

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if Bill had owned the
cab himself and got sued,

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the soccer player who's career was ruined

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would have gotten his
actual million dollars.

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Because Bill set up this corporation

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and put the assets of the cab
company in this corporation,

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he was able to limit his liability.

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The soccer player was only
able to get a $180,000,

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the corporation declared bankruptcy

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but Bill's liability was limited

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to what he had contributed
to the corporation.

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He might say, "My God, this guy over here"

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"his life was somewhat ruined"

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"and he's not able to
kind of get compensated"

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"for I guess how bad his life was ruined."

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Bill is so rich,

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he could probably still make money.

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He should give him the money,

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so he should say this
is kind of an unfair way

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that this corporations are
a way for well off people

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or wealthy people to protect their assets.

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To some degree that's true.

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I want to think about in
this video is the trade-offs

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and really this is just a
balance that society has to pick.

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That this really was on ...

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At least the way I had set
it up in the last video

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was kind of unfair to the soccer player.

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I mean he was only getting a $180,000

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for something that might
have ruined his career.

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If you had talked to Bill,

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Bill might have said well, "That's right,"

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"corporations are kind
of limiting liability"

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"for someone like me."

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He'd say, "Look I
wouldn't have started ..."

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or maybe he wouldn't have
started the cab company

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if he didn't have that limited liability.

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He said, "Look if I had
all of these assets"

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"maybe he had a million dollars."

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Or actually he does
have a million dollars,

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maybe he had a hundred million dollars.

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He said, "Look, if I had
a hundred million dollars"

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"and I didn't have the option
of starting a corporation,"

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"I would have not have
started the cab company"

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"because just for a little
bit of small profit"

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"that I could have
gotten from $80,000 cab"

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"I wouldn't have wanted"

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"to jeopardize the rest of my empire."

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"If I didn't have limited liability,"

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"I wouldn't have started
the business to begin with"

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"and that would not have
been good for society."

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Another argument Bill might make

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and I'm not going to take sides here,

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I'm just saying these are the bounds,

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these are the things
that you have to weigh

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when you set up a legal
code in the society,

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is that Bill said, "Look,
whoever starts a cab business,"

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this cab business that roughly

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you could start with 80, 90, or $100,000.

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They all have the same upside.

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They're taking a certain amount of risk,

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they're providing a service to the city.

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They'll probably make a
similar amount of money.

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He would argue that it would be unfair

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that we have different downsides.

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Maybe Sal right over here,

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maybe my only assets are
the cab plus licenses

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and I have 80K over
there, and the insurance

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which is 1K.

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What Bill would say is, "Look, me and Sal"

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"could both open the
exact same cab business"

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"and if there were no corporations,"

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"if I had to own ..."

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When I say I, I'm talking about Bill.

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If Bill had to own the
assets of the cab company

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or I should say the cab company,

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I should say Bill had to
directly own the assets,

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the cab and licences.

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Even though Bill's and Sal's
upside from the business,

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the profit they could
make would be the same.

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Bill's downside would be more

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because he has ...

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If he didn't have the corporation,

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he would have all of these
other assets at risk.

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Maybe that would make
Bill get a bigger policy

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so that a $100,000 policy,

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maybe he would want to get
a million dollar policy.

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Then a million dollar policy

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will probably cost 10 times more,

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it would cost $10,000 to have that policy.

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If you view it from that way,

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in order to protect his other assets,

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his cost of business would be higher

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just because he happens
to have other things.

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Bill might argue, "Hey
that wouldn't be fair."

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It would again be a reason
why someone with assets

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would be less inclined
to start another business

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which may not be great for society.

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Then the other thing that Bill might argue

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is that look, in exchange
for limited liability

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he might have to be double taxed

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and this is an interesting discussion here

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because you'll probably
hear this term a lot,

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double taxation.

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It's not necessarily going to happen

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but if it's a large corporation,

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the type of corporations
that are publicly traded

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this will happen.

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The idea behind double taxation,

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it's not true for all corporations.

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In fact, let me differentiate it,

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double taxation happens with
a C corporation, a C corp.

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It does not happen with
things called S corps

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and I'm not going to go in detail here

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in what this are.

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S corps, LLC which stands for
limited liability corporation,

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LP which stand for limited partnerships

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or limited liability partnerships, LLPs.

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It doesn't apply to any of these things

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but what happens in double taxation,

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is let's say you have some
corporation, some C corp.

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Let's say that it makes $1,000 in profit.

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It makes $1,000 in profit in a given year.

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Since this is being treated
as kind of a legal entity

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that's kind of like a
person by the government,

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it gets taxed like a person.

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They're going to have to pay some tax.

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Maybe it's let's say 30% tax.

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They're going to have to pay 30% or 35%

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or whatever the corporate tax,

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the marginal corporate tax rate

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for this corporation is going to be.

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They're going to have
to pay $300 in taxes.

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Then let's say this corporation,

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whatever it has left over after taxes

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so this is, let me make
it clear, this is pretax.

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Pretax profit it pays
30% to the government

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and so it has $700 in net, net income.

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This is net of taxes and everything.

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Let's say it then dividends
that out to its shareholders,

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so then it gives it to its shareholders.

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Let's just say for simplicity

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there's one shareholder.

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This is true even if there were
more than one shareholders.

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That one shareholder, he's
getting $700 in income

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based on the fact the he
owns this corporation.

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He is going to have to pay taxes again.

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That's $700, so he has to pay taxes.

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Let me do it this way, let me draw it.

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He has to pay taxes again,

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maybe his marginal tax
rate is another 30%.

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He'll have to pay 30%
to the government again.

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This is 30% on the $700,

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so he'll pay 210.

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$210 in taxes to the government

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based on this net income from his dividend

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that he's getting out
from the C corporation.

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Actually I want to be very careful here,

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I don't want to get into the details

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but dividends can sometimes
be taxed differently

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then straight up income.

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Just for the sake of simplicity

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let's say it's 30%,

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it might be a lower or higher amount.

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This is the idea of double taxation,

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the corporation is used as legal entity,

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it pays taxes, and then after those taxes

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it's able to give its
money back to shareholders

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and then those shareholders
have to pay money

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on that income again.

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That's what the double taxation is.

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Some people say, "Hey
this is horribly unfair."

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"Why should we have to pay taxes twice"

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"on the same money?"

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Then someone would argue, "Well look,"

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"your're getting all of
this limited liability."

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"If you don't like paying
those double taxes"

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"don't set up a corporation"

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"and then just accept the
full straight up liabilities."

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It's a trade off, you
pay the double taxes,

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you get the limited liability.

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Let me be clear here,

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This applies mainly
the large corporations,

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if you want the limited liability

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but you don't want to have
to do this double taxation

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any of these other corporate
structures are available.

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These are usually available to smaller,

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less complicated companies I should say

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than what's going on with the C corp.

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An S corp, you have a limited
number of shareholders.

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LLCs, they can't last forever.

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They stop lasting as soon
as one of the partners

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of the corporation,

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one of the owners of the
corporation pass away

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or declare bankruptcy,

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so there are limitations here.

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These do get the benefits
of limited liability

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and the profits get pass
through to the owners

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without getting taxed first.

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They also have single taxation.

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Going back to the fairness issue,

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if Bill was smart when
he started this company

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he probably would set this up

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as a limited liability corporation

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or as an S corporation

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or something where he doesn't
have to get double taxed

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and still he would avoid having to pay

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the soccer player the
full million dollars.

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Just whatever the company has

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that's all the soccer player would get.

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Let's say the car service becomes huge

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and he wants to make it public one day

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then he would have to
make it a C corporation

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and then he would have to do
the whole double taxation.

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It's something to think about,

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it's interesting to think about,

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is it fair to have something like this?

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The pro-corporation side is

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it encourages entrepreneurship.

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People won't worry about
the downside as much.

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It kind of levels the playing field

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regardless of however
many side assets you have,

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you have the same liability

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for a certain amount of business.

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There's usually certain rules

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that you have to have a
certain amount of money

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in the corporation,

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a certain amount of
insurance in the corporation

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If you don't then possibly

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the court can go after the owner's assets.

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As long as you maintain
it in a responsible,

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well capitalized way, probably won't.

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Then the other trade-off is
this double taxation thing.

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That at least for large corporations,

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the general electrics and the Google's

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and the Microsoft's of the [world]

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they do pay a fee for
this limited liability

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and that's in terms of
this double taxation.

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Although smaller corporations

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right when they're getting started out

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or kind of partnerships,

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they don't have the double taxation

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but they do get the limited liability.


Corporations and limited liability

WEBVTT
Kind: captions
Language: en

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We've all come in contact
with corporations.

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You might work for a corporation.

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You might buy something
from a corporation.

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You might, I don't know, you might own

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shares in a corporation.

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So we kind of have a
sense of what they are,

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but I want to make that sense
a little bit more precise

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in this video and also
give an understanding of

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why corporations even exist.

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And so a corporation, and
you might already kind of

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have a feeling for this,
it's a legal entity.

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It's a legal entity that can
kind of act ... that can act,

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that can act like a
person ... like a person,

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I put the act in quotes
because obviously there's

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some things that a person can
do that a corporation can't.

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A person can smile.

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I guess one can debate
whether a corporation can.

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When I say act, I mean legally.

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So a corporation can sue
another corporation or person.

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A corporation can be sued

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by another corporation or a person.

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A corporation can own things,
just like a person owns things.

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A corporation can owe other
people or corporations

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things, so that's what I mean by "act".

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A corporation can't jump like a
person or laugh like a person.

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Well, that ... maybe that'll be a

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a philosophical discussion
for another video.

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But what I want to focus
on, other than just what a

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corporation is, is why they exist.

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So what's the "why" of a corporation?

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And in my mind, there's two big reasons

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and the first reason and
this is the real biggest one

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is limited liability.

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Sounds like a very fancy word,

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but we'll see it with an example,

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it's not too ... it's not too fancy.

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So there is limited liability
and there's also kind of

00:01:39.604 --> 00:01:41.670
transferrable share ownership.

00:01:41.670 --> 00:01:44.197
Share ownership.

00:01:44.197 --> 00:01:46.070
But what I want to do in
this, which is you know

00:01:46.070 --> 00:01:48.105
essentially more than one
person can own the corporation,

00:01:48.105 --> 00:01:50.336
they can transfer the
shares between people,

00:01:50.336 --> 00:01:51.270
they can trade them.

00:01:51.270 --> 00:01:53.935
But what I want to focus
on is the limited liability

00:01:53.935 --> 00:01:55.778
because even if there's only one person

00:01:55.778 --> 00:01:58.485
who owns the corporation,
if they own the whole thing.

00:01:58.485 --> 00:02:00.668
It still makes sense for them to create it

00:02:00.668 --> 00:02:02.901
because of limited liability.

00:02:02.901 --> 00:02:05.902
And to understand this, let
me give you a little example.

00:02:05.902 --> 00:02:09.485
Let's say that ... Let's, let's
say that there's some guy.

00:02:09.485 --> 00:02:11.369
Let's say he's Bill.

00:02:11.369 --> 00:02:16.401
Let's say it's Bill's ,,,
Bill's assets right over here

00:02:16.401 --> 00:02:19.149
and let's say that he has a house.

00:02:19.149 --> 00:02:24.819
Let's say that he has a
house that is worth $500,000.

00:02:24.819 --> 00:02:29.003
Let's say he has a car,
maybe he has a very nice car.

00:02:29.003 --> 00:02:31.757
Let's say he has a car
that is worth 100 ...

00:02:31.757 --> 00:02:33.021
No, that's too much for a car.

00:02:33.021 --> 00:02:35.117
That's very ... Well,
cars can cost that much

00:02:35.117 --> 00:02:39.201
but let's be ... that's ...
$50,000 is still a nice car.

00:02:39.201 --> 00:02:41.078
Let's say that he has some investments,

00:02:41.078 --> 00:02:44.068
some investment properties
or maybe some stocks

00:02:44.068 --> 00:02:49.069
that are worth another
... another $500,000

00:02:49.069 --> 00:02:51.269
and let's say that he's
interested, you know,

00:02:51.269 --> 00:02:53.484
he looks around his town, he
sees that there's a lot of

00:02:53.484 --> 00:02:55.152
people who could use a ride.

00:02:55.152 --> 00:02:56.271
So to make some extra money,

00:02:56.271 --> 00:02:59.670
he wants to start a cab
or car service business.

00:02:59.670 --> 00:03:01.770
So he goes out there and he doesn't think

00:03:01.770 --> 00:03:05.003
about starting a corporation
and he just goes out there

00:03:05.003 --> 00:03:09.903
and he buys a car, let me
call a cab, plus licenses.

00:03:09.903 --> 00:03:12.335
A cab plus licenses.

00:03:12.335 --> 00:03:15.272
And these things are
actually much more expensive

00:03:15.272 --> 00:03:16.054
than what you would imagine.

00:03:16.054 --> 00:03:18.936
You could ... could talk to a,
the next time you're in a cab,

00:03:18.936 --> 00:03:21.069
it costs much more than
the cost of the car,

00:03:21.069 --> 00:03:23.003
which would probably
be $20,000 or $30,000.

00:03:23.003 --> 00:03:24.818
It comes up close to six figures,

00:03:24.818 --> 00:03:28.305
but let's say that it costs him $80,000.

00:03:28.305 --> 00:03:31.384
And let's say that he spends $1,000,

00:03:31.384 --> 00:03:33.038
at least for this first year.

00:03:33.038 --> 00:03:35.669
Let's say he spends $1,000 on insurance,

00:03:35.669 --> 00:03:38.292
on liability insurance. He realizes,

00:03:38.292 --> 00:03:39.827
"Hey look, maybe you know,

00:03:39.827 --> 00:03:41.823
"maybe something bad happens."

00:03:41.823 --> 00:03:43.819
"This is a cab I could run into somebody"

00:03:43.819 --> 00:03:46.669
"and I get sued. So, I want
to get liability insurance."

00:03:46.669 --> 00:03:48.470
So this is liability insurance,

00:03:48.470 --> 00:03:50.538
in case anything like that happens

00:03:50.538 --> 00:03:54.401
and let's say that it is a ...
Let's say that it is a $100,000

00:03:54.401 --> 00:03:56.396
policy. He says, "You
know, how can I ...",

00:03:56.396 --> 00:03:59.344
you know, he ... he's maybe
a little bit optimistic

00:03:59.344 --> 00:04:00.721
and he thinks that no one can sue him

00:04:00.721 --> 00:04:03.485
for more than $100,000.

00:04:03.485 --> 00:04:09.801
So it's a $100,000 ...
It is a $100,000 policy.

00:04:09.801 --> 00:04:12.804
Now, he goes out there. He
starts driving that cab around,

00:04:12.804 --> 00:04:14.234
picking up people, dropping them off

00:04:14.234 --> 00:04:18.669
and by accident one day, he
rolls over someone's foot

00:04:18.669 --> 00:04:21.534
and that person whose foot he just crushed

00:04:21.534 --> 00:04:25.669
happened to be a ... you know,
a world-class soccer player.

00:04:25.669 --> 00:04:28.735
So he ruined this person's career.

00:04:28.735 --> 00:04:32.735
He ruined the career. So right
here, here's the soccer ...

00:04:32.735 --> 00:04:34.870
the soccer player right over here,

00:04:34.870 --> 00:04:37.069
whose career is ruined or
if you're watching this

00:04:37.069 --> 00:04:40.745
outside of the United
States, a futbol player.

00:04:40.745 --> 00:04:42.735
But anyway, you have this soccer player.

00:04:42.735 --> 00:04:47.388
His crushed ... His foot is crushed.

00:04:47.388 --> 00:04:51.994
And so he sues Bill
because Bill owns the cab.

00:04:51.994 --> 00:04:54.901
And the court ... so he
sues. Let me write it.

00:04:54.901 --> 00:05:02.069
He sues Bill ... sues Bill for $1,000,000.

00:05:02.069 --> 00:05:05.260
$1,000,000 because it ruined his career.

00:05:05.260 --> 00:05:08.153
In fact, that's probably ...
probably deservers more than that

00:05:08.153 --> 00:05:10.336
if he was a world-class player.

00:05:10.336 --> 00:05:12.670
And the court, you know, rightfully says,

00:05:12.670 --> 00:05:14.003
"Look, you crushed this guy's player."

00:05:14.003 --> 00:05:16.069
"You ... you ... you were
kind of ... this guy's foot,"

00:05:16.069 --> 00:05:17.005
"you were kind of negligent."

00:05:17.005 --> 00:05:18.604
"You got a little bit
too close to the curb."

00:05:18.604 --> 00:05:19.735
"His foot was crushed."

00:05:19.735 --> 00:05:22.669
So it rules in the soccer player's favor.

00:05:22.669 --> 00:05:25.484
And so essentially, now Bill's on the line

00:05:25.484 --> 00:05:27.152
for the entire $1,000,000.

00:05:27.152 --> 00:05:29.603
So Bill is going to
have to sell his house.

00:05:29.603 --> 00:05:31.004
He's going to have to see his car.

00:05:31.004 --> 00:05:32.319
Well, he wouldn't have
to sell his car because

00:05:32.319 --> 00:05:34.335
he has investments, sell his investments

00:05:34.335 --> 00:05:37.938
and give his house or the
proceeds from his house

00:05:37.938 --> 00:05:40.819
and investment to the soccer player.

00:05:40.819 --> 00:05:44.734
And so now all Bill is left
with is a car and a cab

00:05:44.734 --> 00:05:48.535
and not much else, so what
did this ... this cab business

00:05:48.535 --> 00:05:50.402
didn't work out too well for Bill.

00:05:50.402 --> 00:05:51.869
It's a small part of his portfolio.

00:05:51.869 --> 00:05:54.003
He didn't even have a chance
to make much money from it

00:05:54.003 --> 00:05:56.234
and it really just wiped him clean.

00:05:56.234 --> 00:05:59.734
Now, what Bill should have
done is create a separate

00:05:59.734 --> 00:06:03.668
corporation, is create
a separate corporation

00:06:03.668 --> 00:06:13.569
called ... called Bill's ... Bill's
Car Service ... Car Service, Inc.

00:06:13.569 --> 00:06:16.495
and transferred enough
money so that the cab,

00:06:16.495 --> 00:06:18.784
the licenseses and the
insurance could be owned

00:06:18.784 --> 00:06:21.388
by this legal entity, by this corporation.

00:06:21.388 --> 00:06:26.670
So here you would have the
cabs ... cab + the license.

00:06:26.670 --> 00:06:28.930
So he would have transferred $80,000

00:06:28.930 --> 00:06:31.332
to the corporation and the
corporation would have bought it

00:06:31.332 --> 00:06:32.537
in the corporation's name.

00:06:32.537 --> 00:06:33.736
Remember, it can act like a person.

00:06:33.736 --> 00:06:35.486
It can buy things in its name.

00:06:35.486 --> 00:06:39.734
The owner of this cab is
Bill's Car Service not Bill.

00:06:39.734 --> 00:06:44.065
It could also get the insurance,
the $100,000 of insurance,

00:06:44.065 --> 00:06:46.483
maybe this is what people
think is a reasonable amount

00:06:46.483 --> 00:06:48.603
for anyone who is running a cab business.

00:06:48.603 --> 00:06:50.070
You have to have at least $100,000.

00:06:50.070 --> 00:06:52.505
The reality is probably it's
a much larger number than that

00:06:52.505 --> 00:06:55.069
and he spends $1,000 to have that policy

00:06:55.069 --> 00:06:58.569
and remember, this is $100,000
of liability insurance.

00:06:58.569 --> 00:07:01.069
Now, let's play out the same situation.

00:07:01.069 --> 00:07:03.402
The cab still rolls over
the soccer player's foot.

00:07:03.402 --> 00:07:05.004
The soccer player's foot gets crushed,

00:07:05.004 --> 00:07:08.870
can't play soccer anymore,
sues whoever owned that cab

00:07:08.870 --> 00:07:10.401
for a $1,000,000.

00:07:10.401 --> 00:07:13.669
But now the owner of that
cab is no longer Bill.

00:07:13.669 --> 00:07:15.802
It's Bill Car Service.

00:07:15.802 --> 00:07:19.686
So he sues this, sues Bill's Car Service

00:07:19.686 --> 00:07:20.937
and let me make it clear.

00:07:20.937 --> 00:07:23.934
In this reality, Bill's assets
wouldn't include the cab,

00:07:23.934 --> 00:07:26.004
the licenses and the insurance anymore.

00:07:26.004 --> 00:07:32.936
It would now include, 100%
ownership ... ownership

00:07:32.936 --> 00:07:38.009
of car service, of Bill's Car Service.

00:07:38.009 --> 00:07:41.819
So the asset is now the shares,
the 100% ownership in this

00:07:41.819 --> 00:07:44.694
but this is a separate legal entity.

00:07:44.694 --> 00:07:46.270
So now the soccer player sues

00:07:46.270 --> 00:07:48.234
the cab service for $1,000,000.

00:07:48.234 --> 00:07:49.669
The court rightfully says,

00:07:49.669 --> 00:07:51.670
"Hey, yeah you owe this dude $1,000,000."

00:07:51.670 --> 00:07:53.423
"You ruined his career.",

00:07:53.438 --> 00:07:55.204
but unfortunate for the player and I guess

00:07:55.204 --> 00:07:57.401
fortunate for Bill, I don't
want to take sides here.

00:07:57.401 --> 00:07:59.027
This is just how reality is set up

00:07:59.027 --> 00:08:01.422
and this is what limited liability is,

00:08:01.422 --> 00:08:05.603
is that now all the car service
can produce is ... Well,

00:08:05.603 --> 00:08:07.936
they could sell ... They
could sell maybe these assets

00:08:07.936 --> 00:08:11.671
for $80,000 and they'll get
$100,000 ... They'll get

00:08:11.671 --> 00:08:14.068
$100,000 from the insurance,

00:08:14.068 --> 00:08:16.153
so all this corporation
will be able to do is

00:08:16.153 --> 00:08:21.670
give the guy $180,000
and then the cooperation

00:08:21.670 --> 00:08:24.038
is just going to go bankrupt.

00:08:24.038 --> 00:08:26.930
So it's going to declare
corporate bankruptcy.

00:08:26.930 --> 00:08:29.370
It's going to go bankrupt.

00:08:29.370 --> 00:08:32.092
So this thing that Bill
owed, this car service

00:08:32.092 --> 00:08:34.272
which maybe originally he could
write on his books as being

00:08:34.302 --> 00:08:38.700
worth $81,000, had insurance
policy and all this other stuff.

00:08:38.700 --> 00:08:39.988
It's now worth 0.

00:08:39.988 --> 00:08:43.021
The corporation's worth 0,

00:08:43.021 --> 00:08:46.555
but the soccer player can't
go after anything else.

00:08:46.555 --> 00:08:47.730
We'll talk about in the future,

00:08:47.730 --> 00:08:49.530
of reasons why they can.

00:08:49.530 --> 00:08:51.188
If he didn't have enough insurance

00:08:51.188 --> 00:08:52.771
or if he didn't ... if he was running this

00:08:52.771 --> 00:08:54.197
like his own personal business.

00:08:54.197 --> 00:08:56.864
If this car service didn't
have its own bank account

00:08:56.864 --> 00:08:58.995
or if it didn't ... if
it was undercapitalized,

00:08:58.995 --> 00:09:00.972
if it didn't have enough money in it,

00:09:00.972 --> 00:09:04.518
then maybe the court would
let the soccer player or

00:09:04.518 --> 00:09:07.020
kind of "pierce the
corporate veil" they say,

00:09:07.020 --> 00:09:09.397
which is a very fancy
way of still going after

00:09:09.397 --> 00:09:11.129
the owners of the corporation.

00:09:11.129 --> 00:09:14.272
But as long as this corporation
was reasonably capitalized,

00:09:14.272 --> 00:09:15.938
and I'm just made up these numbers here,

00:09:15.938 --> 00:09:20.354
the soccer player will only be
able to go after this corporation.

00:09:20.354 --> 00:09:24.798
So what you see here is that
Bill's liability was limited.

00:09:24.798 --> 00:09:26.688
This was limited liability.

00:09:26.688 --> 00:09:27.938
The most he could lose

00:09:27.938 --> 00:09:30.463
as long as he ran the car service properly

00:09:30.463 --> 00:09:32.798
and had enough insurance
and had enough money in the

00:09:32.798 --> 00:09:34.795
actual organization.

00:09:34.795 --> 00:09:37.063
As long as he ran it in the right way,

00:09:37.063 --> 00:09:41.211
the most he could lose is the
value of the organization.

00:09:41.226 --> 00:09:43.893
If this organization,
you know, hits somebody

00:09:43.893 --> 00:09:45.728
or rolled over their foot,

00:09:45.728 --> 00:09:48.198
and ... but he ran the
organization properly,

00:09:48.198 --> 00:09:50.864
they can;t go over the rest of his assets.

00:09:50.864 --> 00:09:52.789
So this is the number one reason

00:09:52.804 --> 00:09:56.804
why people have corporations,
for the limited liability.


Term life insurance and death probability

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.110
I'm thinking about getting
life insurance because I

00:00:03.110 --> 00:00:05.610
have a mortgage and
I have a young son

00:00:05.610 --> 00:00:07.644
and another baby on the way.

00:00:07.644 --> 00:00:09.310
And so if anything
were to happen to me,

00:00:09.310 --> 00:00:12.380
I'd want them to at least be
able to pay off the mortgage

00:00:12.380 --> 00:00:14.890
and then maybe have some
money left over for college

00:00:14.890 --> 00:00:17.405
and to live, and whatever else.

00:00:17.405 --> 00:00:19.030
And so I went to the
insurance company,

00:00:19.030 --> 00:00:20.905
and I said I want to
get a $1 million policy.

00:00:20.905 --> 00:00:24.080
And what I'm actually
getting a quote on

00:00:24.080 --> 00:00:26.380
is a term life policy,
which is really--

00:00:26.380 --> 00:00:27.880
I just care about
the next 20 years.

00:00:27.880 --> 00:00:30.700
After those 20 years, hopefully,
I can pay off my mortgage.

00:00:30.700 --> 00:00:32.290
There'll be money saved up.

00:00:32.290 --> 00:00:34.300
Hopefully, my kids
would kind of at least

00:00:34.300 --> 00:00:36.300
have maybe gotten
to college or I

00:00:36.300 --> 00:00:38.215
would have saved up
enough money for college.

00:00:38.215 --> 00:00:40.340
So that's why I'm willing
to do a term life policy.

00:00:40.340 --> 00:00:43.000
The other option is to do
a whole life policy, where

00:00:43.000 --> 00:00:45.310
you could pay a
certain amount per year

00:00:45.310 --> 00:00:46.760
for the rest of your life.

00:00:46.760 --> 00:00:50.100
At any point you die, you
get the million dollars.

00:00:50.100 --> 00:00:53.600
In a term life, I'm only
going to pay a $500 per year

00:00:53.600 --> 00:00:55.620
for the next 20 years.

00:00:55.620 --> 00:00:57.740
If at any point over
those 20 years I die,

00:00:57.740 --> 00:00:59.440
my family gets a million.

00:00:59.440 --> 00:01:02.255
At the 21st year, I have
to get a new policy.

00:01:02.255 --> 00:01:03.880
And since I'm going
to be older and I'd

00:01:03.880 --> 00:01:06.190
have a higher chance
of dying at that point,

00:01:06.190 --> 00:01:08.972
then it's probably going
to be more expensive for me

00:01:08.972 --> 00:01:09.680
to get insurance.

00:01:09.680 --> 00:01:12.160
But I really am just worried
about the next 20 years.

00:01:12.160 --> 00:01:13.618
But what I want to
do in this video

00:01:13.618 --> 00:01:15.340
is think about given
these numbers that

00:01:15.340 --> 00:01:17.940
have been quoted to me by
the insurance company, what

00:01:17.940 --> 00:01:20.780
do they think that
my odds of dying

00:01:20.780 --> 00:01:23.941
are over the next 20 years?

00:01:23.941 --> 00:01:25.940
So what I want to think
about is the probability

00:01:25.940 --> 00:01:33.900
of Sal's death in
20 years, based

00:01:33.900 --> 00:01:35.720
on what the people at
the insurance company

00:01:35.720 --> 00:01:36.344
are telling me.

00:01:36.344 --> 00:01:38.610
Or at least, what's
the maximum probability

00:01:38.610 --> 00:01:41.570
of my death in order
for them to make money?

00:01:41.570 --> 00:01:44.070
And the way to think about it,
or one way to think about it,

00:01:44.070 --> 00:01:45.620
kind of a
back-of-the-envelope way,

00:01:45.620 --> 00:01:48.220
is to think about what's the
total premiums they're getting

00:01:48.220 --> 00:01:51.660
over the life of this
policy divided by how much

00:01:51.660 --> 00:01:52.910
they're insuring me for.

00:01:52.910 --> 00:02:00.370
So they're getting $500
times 20 years is equal to,

00:02:00.370 --> 00:02:05.060
that's $10,000 over the
life of this policy.

00:02:05.060 --> 00:02:08.335
And they are insuring
me for $1 million.

00:02:13.050 --> 00:02:15.210
So they're getting-- let's
see those 0s cancel out,

00:02:15.210 --> 00:02:17.370
this 0 cancels out--
they're getting,

00:02:17.370 --> 00:02:24.030
over the life of the
policy, $1 in premiums

00:02:24.030 --> 00:02:33.090
for every $100 in insurance.

00:02:33.090 --> 00:02:34.920
Or another way to
think about it.

00:02:34.920 --> 00:02:39.010
Let's say that there were 100
Sals, 100 34-year-olds looking

00:02:39.010 --> 00:02:41.680
to get 20-year term
life insurance.

00:02:41.680 --> 00:02:43.310
And they insured all of them.

00:02:43.310 --> 00:02:48.250
So if you multiplied
this times 100,

00:02:48.250 --> 00:02:49.965
they would get $100 in premiums.

00:02:54.370 --> 00:02:56.660
This is the case where
you have 100 Sals,

00:02:56.660 --> 00:02:58.770
or 100 people who are
pretty similar to me.

00:02:58.770 --> 00:03:01.560
100 Sals.

00:03:01.560 --> 00:03:03.450
They would get $100 in premium.

00:03:03.450 --> 00:03:09.150
And the only way that they
could make money is if, at most,

00:03:09.150 --> 00:03:12.280
one of those Sals-- or really
just break even-- if, at most,

00:03:12.280 --> 00:03:14.950
1 of those Sals were to die.

00:03:14.950 --> 00:03:23.987
So break even if
only 1 Sal dies.

00:03:23.987 --> 00:03:25.320
I don't like talking about this.

00:03:25.320 --> 00:03:26.650
It's a little bit morbid.

00:03:26.650 --> 00:03:28.191
So one way to think
about it, they're

00:03:28.191 --> 00:03:29.860
getting $1 premium
for $100 insurance.

00:03:29.860 --> 00:03:33.090
Or if they had 100 Sals, they
would get $100 in premium,

00:03:33.090 --> 00:03:36.050
and the only way they
would break even, if only 1

00:03:36.050 --> 00:03:38.010
of those Sal dies.

00:03:38.010 --> 00:03:40.190
So what they're really
saying is that the only way

00:03:40.190 --> 00:03:43.840
they can break even is if
the probability of Sal dying

00:03:43.840 --> 00:03:51.525
in the next 20 years is less
than or equal to 1 in 100.

00:03:51.525 --> 00:03:52.900
And this is an
insurance company.

00:03:52.900 --> 00:03:54.880
They're trying to make money.

00:03:54.880 --> 00:03:56.860
So they're probably
giving these numbers

00:03:56.860 --> 00:03:59.230
because they think the
probability of me dying

00:03:59.230 --> 00:04:02.140
is a good-- maybe it's 1
in 200 or it's 1 in 300.

00:04:02.140 --> 00:04:04.790
Something lower, so
that they can insure--

00:04:04.790 --> 00:04:08.790
one way to think about it--
they could insure more Sals

00:04:08.790 --> 00:04:10.992
for every $100 in premium
they have to pay out.

00:04:10.992 --> 00:04:12.950
But either way, it's a
back-of-the-envelope way

00:04:12.950 --> 00:04:14.110
of thinking about it.

00:04:14.110 --> 00:04:16.940
And it actually makes me
feel a little bit better

00:04:16.940 --> 00:04:21.900
because 1 in 100 over the
next 20 years isn't too bad.


Back-of-envelope office space conundrum

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:02.360
What I want to do
in this video is

00:00:02.360 --> 00:00:05.430
kind of give a back
of the envelope way

00:00:05.430 --> 00:00:07.610
for thinking about a
conundrum that the Khan

00:00:07.610 --> 00:00:09.040
Academy is facing right now.

00:00:09.040 --> 00:00:10.456
And you could do
a little bit more

00:00:10.456 --> 00:00:12.930
of a precise way with present
valuing and all of that.

00:00:12.930 --> 00:00:14.680
But this will give you
a pretty good sense

00:00:14.680 --> 00:00:15.949
of how to make this decision.

00:00:15.949 --> 00:00:17.740
So we're seriously
looking at office space.

00:00:17.740 --> 00:00:20.600
I've changed these numbers to
simplify things a little bit.

00:00:20.600 --> 00:00:22.760
But there's a nice
office, and it's

00:00:22.760 --> 00:00:25.862
going for $2,000 per month.

00:00:25.862 --> 00:00:27.320
And then there's
this dingy office.

00:00:27.320 --> 00:00:28.470
It's really too dingy.

00:00:28.470 --> 00:00:30.636
It's a converted nightclub
and it's really too dingy

00:00:30.636 --> 00:00:35.810
to be a respectable place to
work, but it's going for $1,000

00:00:35.810 --> 00:00:36.630
a month.

00:00:36.630 --> 00:00:37.980
And I talked to a contractor.

00:00:37.980 --> 00:00:39.897
And once again, I'm
simplifying these numbers.

00:00:39.897 --> 00:00:41.521
These numbers aren't
the exact numbers,

00:00:41.521 --> 00:00:43.410
but it makes the math
a little bit simpler.

00:00:43.410 --> 00:00:48.400
The contractor says it
would cost us about $10,000

00:00:48.400 --> 00:00:53.290
to get this dingy office to the
condition of the nice office.

00:00:53.290 --> 00:00:59.360
And in either situation, we
are thinking about a 24 month

00:00:59.360 --> 00:01:00.960
lease, so a two year lease.

00:01:04.040 --> 00:01:08.860
And the contractor said it
would take one month of work

00:01:08.860 --> 00:01:11.650
to get the dingy office
space up to speed.

00:01:11.650 --> 00:01:15.470
So the nice office
space landlord he says,

00:01:15.470 --> 00:01:17.239
look, it'll be a 24 month lease.

00:01:17.239 --> 00:01:18.280
You could just occupy it.

00:01:18.280 --> 00:01:19.370
It's ready to go.

00:01:19.370 --> 00:01:21.010
The dingy office
space, I talked him

00:01:21.010 --> 00:01:25.250
about this idea of us investing
money in repairing it.

00:01:25.250 --> 00:01:27.660
He's like, OK, if you
need to repair it--

00:01:27.660 --> 00:01:30.600
he's kind not the easiest
person to deal with-- he says,

00:01:30.600 --> 00:01:32.730
I'm going to charge you
an extra month of rent.

00:01:32.730 --> 00:01:35.260
So even while I'm repairing
it, I have to pay rent.

00:01:35.260 --> 00:01:38.380
So he'll say, I'm willing
to do a 25 month contract,

00:01:38.380 --> 00:01:42.420
a 25 month lease with you.

00:01:42.420 --> 00:01:44.740
So you could still, once
you repair it, you still

00:01:44.740 --> 00:01:48.569
get two years of service
from the dingy office.

00:01:48.569 --> 00:01:49.610
So which one should I do?

00:01:49.610 --> 00:01:51.660
And I'll tell you
my emotions, I want

00:01:51.660 --> 00:01:54.360
to do the nice one because I
don't want to spend $10,000

00:01:54.360 --> 00:01:57.770
and make this all nice and then
two years later, the landlord

00:01:57.770 --> 00:01:59.212
could raise the rent.

00:01:59.212 --> 00:02:00.920
But let's just think
about it rationally.

00:02:00.920 --> 00:02:02.055
Which one makes sense?

00:02:02.055 --> 00:02:03.680
And the easiest way
to think about that

00:02:03.680 --> 00:02:06.540
is, what is the
effective amount of rent,

00:02:06.540 --> 00:02:08.150
in a very back of
the envelope way,

00:02:08.150 --> 00:02:10.639
that we're paying
for the dingy office?

00:02:10.639 --> 00:02:17.690
So if you think about it
over the entire lease,

00:02:17.690 --> 00:02:19.590
we're going to have 25 months.

00:02:19.590 --> 00:02:21.900
We're going to pay
$1,000 per month.

00:02:21.900 --> 00:02:27.850
So it's going to be $25,000
plus $10,000 to repair it.

00:02:27.850 --> 00:02:30.110
So plus $10,000.

00:02:30.110 --> 00:02:32.300
I'll just write 10k
like I did before.

00:02:32.300 --> 00:02:33.780
Plus 10k.

00:02:33.780 --> 00:02:36.347
And in terms of the useful
life, because over here

00:02:36.347 --> 00:02:38.680
when we're saying per month,
every month here is useful.

00:02:38.680 --> 00:02:40.259
So how many useful
months are here?

00:02:40.259 --> 00:02:42.300
Well we're going to spend
one month to repair it.

00:02:42.300 --> 00:02:44.660
We only get 24 useful months.

00:02:44.660 --> 00:02:49.340
So we only get 24 useful months.

00:02:49.340 --> 00:02:51.860
So if you view it this way,
not doing any fancy present

00:02:51.860 --> 00:02:53.810
valuing, just back
of the envelope,

00:02:53.810 --> 00:03:02.730
it is $35,000 over 24 months.

00:03:02.730 --> 00:03:05.219
And so let's get
our calculator out.

00:03:05.219 --> 00:03:07.010
I guess it's not pure
back of the envelope,

00:03:07.010 --> 00:03:10.590
but pretty close to pure
back of the envelope.

00:03:10.590 --> 00:03:18.680
35,000 divided by 24
is equal to 1,458.

00:03:18.680 --> 00:03:26.430
So this is equal to
approximately $1,458 per month.

00:03:26.430 --> 00:03:29.664
So even though I don't want
to invest in this property,

00:03:29.664 --> 00:03:31.580
and I'm only going to
be able to use two years

00:03:31.580 --> 00:03:33.371
and this person's going
to be able to raise

00:03:33.371 --> 00:03:37.500
the rent after that, it actually
still makes a lot of sense.

00:03:37.500 --> 00:03:40.335
It still makes actually
almost $500 a month of sense

00:03:40.335 --> 00:03:42.200
or more than $500
a month of sense

00:03:42.200 --> 00:03:44.290
in this kind of back
of the envelope way

00:03:44.290 --> 00:03:48.850
to get the dingy
office and repair it.


Recognizing odd and even functions

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.670
Let's see if we can learn
a thing or two about even

00:00:04.670 --> 00:00:06.940
functions and odd functions.

00:00:06.940 --> 00:00:10.210
So even functions, and on the
right-hand side over here,

00:00:10.210 --> 00:00:12.210
we'll talk about odd functions.

00:00:12.210 --> 00:00:14.790
If we have time, we'll
talk about functions

00:00:14.790 --> 00:00:18.230
that are neither even nor odd.

00:00:18.230 --> 00:00:21.001
So before I go into kind
of a formal definition

00:00:21.001 --> 00:00:22.750
of even functions, I
just want to show you

00:00:22.750 --> 00:00:25.305
what they look like visually,
because I think that's probably

00:00:25.305 --> 00:00:26.830
the easiest way
to recognize them.

00:00:26.830 --> 00:00:29.260
And then it'll also make
a little bit more sense

00:00:29.260 --> 00:00:32.670
when we talk about the formal
definition of an even function.

00:00:32.670 --> 00:00:37.490
So let me draw some
coordinate axes here, x-axis.

00:00:37.490 --> 00:00:39.940
And then, let me
see if I can draw

00:00:39.940 --> 00:00:41.630
that a little bit straighter.

00:00:41.630 --> 00:00:44.720
This right over
here is my y-axis,

00:00:44.720 --> 00:00:49.060
or I could say y is equal to
f of x axis, just like that.

00:00:49.060 --> 00:00:53.700
And then let me draw
the graph of f of x.

00:00:53.700 --> 00:00:56.720
f of x is equal
to x-squared, or y

00:00:56.720 --> 00:00:58.690
is equal to
x-squared, either one.

00:00:58.690 --> 00:00:59.650
So let me draw it.

00:00:59.650 --> 00:01:02.020
In the first quadrant,
it looks like this.

00:01:02.020 --> 00:01:07.140
And then in the second
quadrant, it looks like this.

00:01:07.140 --> 00:01:11.810
It looks like-- oh, let me try
to draw it so it's symmetric.

00:01:11.810 --> 00:01:13.800
Pretty good job.

00:01:13.800 --> 00:01:19.486
The f of x is equal to x
squared is an even function.

00:01:19.486 --> 00:01:20.860
And the way that
you recognize it

00:01:20.860 --> 00:01:25.440
is because it has this
symmetry around the y-axis.

00:01:25.440 --> 00:01:28.800
If you take what's going
on on the right-hand side,

00:01:28.800 --> 00:01:31.680
to the right of the y-axis,
and you just reflect it

00:01:31.680 --> 00:01:35.180
over the y-axis, you get the
other side of the function.

00:01:35.180 --> 00:01:38.940
And that's what tells you
it is an even function.

00:01:38.940 --> 00:01:41.700
And I want to show you one
interesting property here.

00:01:41.700 --> 00:01:43.570
If you take any
x-value-- let's say

00:01:43.570 --> 00:01:44.940
you take a positive x-value.

00:01:44.940 --> 00:01:48.270
Let's say you take the
value, x is equal to 2.

00:01:48.270 --> 00:01:51.220
If you find f of 2,
that's going to be 4.

00:01:51.220 --> 00:01:54.873
That's going to be 4 for this
particular function for f

00:01:54.873 --> 00:01:56.250
of x is equal to x squared.

00:01:56.250 --> 00:01:57.920
2 squared is equal to 4.

00:01:57.920 --> 00:02:00.340
And if you took the
negative version of 2--

00:02:00.340 --> 00:02:05.760
so if you took negative 2,
and you evaluated the function

00:02:05.760 --> 00:02:10.520
there, you are also
going to get 4.

00:02:10.520 --> 00:02:13.820
And this, hopefully, or maybe
makes complete sense to you.

00:02:13.820 --> 00:02:16.125
You're like, well, Sal,
obviously if I just

00:02:16.125 --> 00:02:18.810
reflect this function
over the y-axis, that's

00:02:18.810 --> 00:02:19.700
going to be the case.

00:02:19.700 --> 00:02:24.244
Whatever function value I get at
the positive value of a number,

00:02:24.244 --> 00:02:25.660
I'm going to get
the same function

00:02:25.660 --> 00:02:27.260
value at the negative value.

00:02:27.260 --> 00:02:31.450
And this is what kind of leads
us to the formal definition.

00:02:31.450 --> 00:02:34.970
If a function is even--
or I could say a function

00:02:34.970 --> 00:02:37.940
is even if and only
of-- so it's even.

00:02:37.940 --> 00:02:41.450
And don't get confused
between the term even function

00:02:41.450 --> 00:02:42.760
and the term even number.

00:02:42.760 --> 00:02:45.480
They're completely
different kind of ideas.

00:02:45.480 --> 00:02:47.680
So there's not, at least
an obvious connection

00:02:47.680 --> 00:02:50.270
that I know of, between even
functions and even numbers

00:02:50.270 --> 00:02:52.450
or odd functions
and odd numbers.

00:02:52.450 --> 00:02:56.500
So you're an even
function if and only

00:02:56.500 --> 00:03:03.570
if, f of x is equal
to f of negative x.

00:03:03.570 --> 00:03:05.620
And the reason why I
didn't introduce this

00:03:05.620 --> 00:03:06.780
from the beginning--
because this is really

00:03:06.780 --> 00:03:08.030
the definition of
an even function--

00:03:08.030 --> 00:03:10.655
is when you look at this, you're
like hey, what does this mean?

00:03:10.655 --> 00:03:12.340
f of x is equal to
f of negative x.

00:03:12.340 --> 00:03:14.250
And all it does mean is this.

00:03:14.250 --> 00:03:17.990
It means that if I were to
take f of 2, f of 2 is 4.

00:03:17.990 --> 00:03:20.610
So let me show you with
a particular case. f of 2

00:03:20.610 --> 00:03:25.230
is equal to f of negative 2.

00:03:25.230 --> 00:03:28.615
And this particular case for
f of x is equal to x squared,

00:03:28.615 --> 00:03:32.740
they are both equal to 4.

00:03:32.740 --> 00:03:35.220
So really, this is
just another way

00:03:35.220 --> 00:03:38.010
of saying that the
function can be reflected,

00:03:38.010 --> 00:03:40.870
or the left side of the
function is the reflection

00:03:40.870 --> 00:03:42.750
of the right side
of the function

00:03:42.750 --> 00:03:47.410
across the vertical
axis, across the y-axis.

00:03:47.410 --> 00:03:50.040
Now just to make sure we have
a decent understanding here,

00:03:50.040 --> 00:03:53.190
let me draw a few
more even functions.

00:03:53.190 --> 00:03:57.030
And I'm going to draw some
fairly wacky things just so you

00:03:57.030 --> 00:03:59.770
really kind of learn to
visually recognize them.

00:03:59.770 --> 00:04:08.510
So let's say a
function like this,

00:04:08.510 --> 00:04:13.176
it maybe jumps up
to here, and maybe

00:04:13.176 --> 00:04:15.250
it does something like that.

00:04:15.250 --> 00:04:17.240
And then on this side,
it does the same thing.

00:04:17.240 --> 00:04:20.310
It's the reflection,
so it jumps up here,

00:04:20.310 --> 00:04:23.420
then it goes like this,
then it goes like this.

00:04:23.420 --> 00:04:26.810
I'm trying to draw it so it's
the mirror image of each other.

00:04:26.810 --> 00:04:29.280
This is an even function.

00:04:29.280 --> 00:04:31.440
You take what's going
on on the right hand

00:04:31.440 --> 00:04:34.150
side of this function and
you literally just reflect it

00:04:34.150 --> 00:04:38.046
over the y-axis, and you get the
left hand side of the function.

00:04:38.046 --> 00:04:39.670
And you could see
that even this holds.

00:04:39.670 --> 00:04:45.320
If I take some value-- let's say
that this value right here is,

00:04:45.320 --> 00:04:46.810
I don't know, 3.

00:04:46.810 --> 00:04:52.720
And let's say that f of 3 over
here is equal to, let's say,

00:04:52.720 --> 00:04:54.000
that that is 5.

00:04:54.000 --> 00:04:55.160
So this is 5.

00:04:58.170 --> 00:05:02.340
We see that f of negative 3 is
also going to be equal to 5.

00:05:02.340 --> 00:05:06.030
And that's what our definition
of an even function told us.

00:05:06.030 --> 00:05:09.860
And I can draw, let me just draw
one more to really make sure.

00:05:09.860 --> 00:05:12.780
I'll do the axis in
that same green color.

00:05:12.780 --> 00:05:16.750
Let me do one more like this.

00:05:16.750 --> 00:05:21.280
And you could have maybe some
type of trigonometric looking

00:05:21.280 --> 00:05:27.380
function that looks like
this, that looks like that.

00:05:27.380 --> 00:05:30.280
And it keeps going
in either direction.

00:05:30.280 --> 00:05:32.950
So something like this
would also be even.

00:05:32.950 --> 00:05:34.510
So all of these
are even functions.

00:05:34.510 --> 00:05:38.460
Now, you are probably thinking,
well, what is an odd function?

00:05:38.460 --> 00:05:41.590
And let me draw an
odd function for you.

00:05:41.590 --> 00:05:45.110
So let me draw the
axis once again.

00:05:45.110 --> 00:05:49.650
x-axis, y-axis, or
the f of x-axis.

00:05:49.650 --> 00:05:52.620
And to show you an
odd function, I'll

00:05:52.620 --> 00:05:55.040
give you a particular odd
function, maybe the most famous

00:05:55.040 --> 00:05:56.190
of the odd functions.

00:05:56.190 --> 00:05:58.870
This is probably the most
famous of the even functions.

00:05:58.870 --> 00:06:01.820
And it is f of x--
although there are probably

00:06:01.820 --> 00:06:04.160
other contenders for
most famous odd function.

00:06:04.160 --> 00:06:07.440
f of x is equal to x
to the third power.

00:06:07.440 --> 00:06:10.350
And it looks like-- you might
have seen the graph of it.

00:06:10.350 --> 00:06:13.690
If you haven't, you can graph
it by trying some points.

00:06:13.690 --> 00:06:15.650
It looks like that.

00:06:15.650 --> 00:06:19.140
And the way to visually
recognize an odd function

00:06:19.140 --> 00:06:22.945
is you look at what's going
on to the right of the y-axis.

00:06:22.945 --> 00:06:25.070
Once again, this is the
y-axis, this is the x-axis.

00:06:25.070 --> 00:06:27.970
You have all of this business
to the right of the y-axis.

00:06:27.970 --> 00:06:30.330
If you reflect it
over the y-axis,

00:06:30.330 --> 00:06:33.605
you would get
something like this.

00:06:33.605 --> 00:06:35.980
And if the left side of this
graph looked like this, then

00:06:35.980 --> 00:06:37.730
we would be dealing
with an even function.

00:06:37.730 --> 00:06:38.940
Clearly it doesn't.

00:06:38.940 --> 00:06:42.520
To make this an odd function, we
reflect it once over the y-axis

00:06:42.520 --> 00:06:44.750
and then reflect
it over the x-axis.

00:06:44.750 --> 00:06:46.230
Or another way to
think about it,

00:06:46.230 --> 00:06:49.714
reflect it once over the y-axis
and then make it negative.

00:06:49.714 --> 00:06:51.130
Either way, it
will get you there.

00:06:51.130 --> 00:06:53.940
Or you could even reflect
it over the x-axis and then

00:06:53.940 --> 00:06:56.250
the y-axis, so you are kind
of doing two reflections.

00:06:56.250 --> 00:06:58.020
And so clearly if
you take this up here

00:06:58.020 --> 00:07:02.520
and then you reflect it over the
x-axis, you get these values,

00:07:02.520 --> 00:07:05.510
you get this part of the
graph right over here.

00:07:05.510 --> 00:07:08.750
And if you try to do it
with a particular point,

00:07:08.750 --> 00:07:11.990
and I'm doing this to kind of
hint at what the definition,

00:07:11.990 --> 00:07:14.860
the formal definition of an
odd function is going to be.

00:07:14.860 --> 00:07:18.130
Let's try a point,
let's try 2 again.

00:07:18.130 --> 00:07:21.070
If you had the point
2, f of 2 is 8.

00:07:21.070 --> 00:07:24.780
So f of 2 is equal to 8.

00:07:24.780 --> 00:07:28.820
Now what happens if
we take negative 2?

00:07:28.820 --> 00:07:32.830
f of negative 2, negative
2 to the third power,

00:07:32.830 --> 00:07:34.710
that's just going
to be negative 8.

00:07:34.710 --> 00:07:38.970
So f of negative 2 is
equal to negative 8.

00:07:38.970 --> 00:07:41.780
And in general, if we take-- so
let me just write it over here.

00:07:41.780 --> 00:07:44.730
f of 2-- so we're just
doing one particular example

00:07:44.730 --> 00:07:47.510
from this particular function.

00:07:47.510 --> 00:07:53.240
We have f of 2 is equal
to, not f of negative 2.

00:07:53.240 --> 00:07:55.010
8 does not equal negative 8.

00:07:55.010 --> 00:07:57.965
8 is equal to the
negative of negative 8

00:07:57.965 --> 00:07:59.620
because that's positive 8.

00:07:59.620 --> 00:08:05.446
So f of 2 is equal to the
negative of f of negative 2.

00:08:05.446 --> 00:08:07.320
We figured out-- just
want to make it clear--

00:08:07.320 --> 00:08:09.240
we figured out f of 2 is 8.

00:08:09.240 --> 00:08:10.690
2 to the third power is 8.

00:08:10.690 --> 00:08:14.220
We know that f of
negative 2 is negative 8.

00:08:14.220 --> 00:08:16.080
Negative 2 to the third
power is negative 8.

00:08:16.080 --> 00:08:18.550
So you have the
negative of negative 8,

00:08:18.550 --> 00:08:20.690
negatives cancel out,
and it works out.

00:08:20.690 --> 00:08:23.110
So in general, you
have an odd function.

00:08:23.110 --> 00:08:25.070
So here's the definition.

00:08:25.070 --> 00:08:27.440
You are dealing
with an odd function

00:08:27.440 --> 00:08:32.190
if and only if f of
x for all the x's

00:08:32.190 --> 00:08:35.289
that are defined on that
function, or for which

00:08:35.289 --> 00:08:37.730
that function is
defined, if f of x

00:08:37.730 --> 00:08:41.104
is equal to the negative
of f of negative x.

00:08:41.104 --> 00:08:42.770
Or you'll sometimes
see it the other way

00:08:42.770 --> 00:08:46.880
if you multiply both sides of
this equation by negative 1,

00:08:46.880 --> 00:08:51.946
you would get negative f of x
is equal to f of negative x.

00:08:51.946 --> 00:08:53.320
And sometimes
you'll see it where

00:08:53.320 --> 00:08:56.200
it's swapped around where
they'll say f of negative x

00:08:56.200 --> 00:09:02.900
is equal to-- let me
write that careful--

00:09:02.900 --> 00:09:06.310
is equal to negative f of x.

00:09:06.310 --> 00:09:08.540
I just swapped these two sides.

00:09:08.540 --> 00:09:12.160
So let me just draw you
some more odd functions.

00:09:15.020 --> 00:09:17.600
So I'll do these visually.

00:09:17.600 --> 00:09:21.950
So let me draw that
a little bit cleaner.

00:09:21.950 --> 00:09:27.350
So if you have
maybe the function

00:09:27.350 --> 00:09:33.760
does something wacky like
this on the right hand side.

00:09:33.760 --> 00:09:36.090
If it was even, you
would reflect it there.

00:09:36.090 --> 00:09:37.590
But we want to have
an odd function,

00:09:37.590 --> 00:09:39.110
so we're going to
reflect it again.

00:09:39.110 --> 00:09:41.700
So the rest of the function
is going to look like this.

00:09:41.700 --> 00:09:43.930
So what I've drawn in
the non-dotted lines,

00:09:43.930 --> 00:09:45.941
this right here is
an odd function.

00:09:45.941 --> 00:09:47.690
And you could even
look at the definition.

00:09:47.690 --> 00:09:51.310
If you take some
value, a, and then

00:09:51.310 --> 00:09:53.820
you take f of a, which
would put you up here.

00:09:53.820 --> 00:09:55.260
This right here would be f of a.

00:09:55.260 --> 00:09:56.910
If you take the
negative value of that,

00:09:56.910 --> 00:10:00.580
if you took negative a
here, f of negative a

00:10:00.580 --> 00:10:02.100
is going to be down here.

00:10:02.100 --> 00:10:07.200
So f of negative
a, it's going to be

00:10:07.200 --> 00:10:09.110
the same distance from
the horizontal axis.

00:10:09.110 --> 00:10:11.740
It's not completely clear
the way I drew it just now.

00:10:11.740 --> 00:10:14.200
So it's maybe going to
be like right over here.

00:10:14.200 --> 00:10:18.230
So this right over here is going
to be f of negative a, which

00:10:18.230 --> 00:10:21.160
is the same distance from
the origin as f of a,

00:10:21.160 --> 00:10:22.710
it's just the negative.

00:10:22.710 --> 00:10:24.570
I didn't completely
draw it to scale.

00:10:24.570 --> 00:10:28.020
Let me draw one more
of these odd functions.

00:10:28.020 --> 00:10:29.470
I think you might get the point.

00:10:29.470 --> 00:10:31.190
Actually, I'll draw a very
simple odd function, just

00:10:31.190 --> 00:10:32.606
to show you that
it doesn't always

00:10:32.606 --> 00:10:34.220
have to be something crazy.

00:10:34.220 --> 00:10:37.650
So a very simple
odd function would

00:10:37.650 --> 00:10:44.520
be y is equal to x,
something like this.

00:10:44.520 --> 00:10:45.800
Whoops.

00:10:45.800 --> 00:10:49.490
y is equal going
through the origin.

00:10:49.490 --> 00:10:51.669
You reflect what's on the
right onto to the left.

00:10:51.669 --> 00:10:52.210
You get that.

00:10:52.210 --> 00:10:53.585
And then you
reflect it down, you

00:10:53.585 --> 00:10:55.540
get all of this stuff
in the third quadrant.

00:10:55.540 --> 00:10:57.900
So this is also an odd function.

00:10:57.900 --> 00:11:00.260
Now, I want to leave
you with a few things

00:11:00.260 --> 00:11:02.570
that are not odd functions
and that sometimes might

00:11:02.570 --> 00:11:05.590
be confused to be odd functions.

00:11:05.590 --> 00:11:09.340
So you might have
something like this

00:11:09.340 --> 00:11:13.430
where maybe you have a
parabola, but it's not

00:11:13.430 --> 00:11:14.830
symmetric around the y-axis.

00:11:14.830 --> 00:11:16.496
And your temptation
might be, hey, there

00:11:16.496 --> 00:11:18.070
is this symmetry
for this parabola.

00:11:18.070 --> 00:11:22.600
But it's not being
reflected around the y-axis.

00:11:22.600 --> 00:11:24.880
You don't have a situation
here where f of x

00:11:24.880 --> 00:11:27.830
is equal to f of negative x.

00:11:27.830 --> 00:11:32.600
So this is neither odd nor even.

00:11:32.600 --> 00:11:35.160
Similarly, you
might see, let's say

00:11:35.160 --> 00:11:39.090
you see a shifted
cubic function.

00:11:39.090 --> 00:11:41.790
So say you have
something like this.

00:11:41.790 --> 00:11:46.350
Let's say you have x
to the third plus 1.

00:11:46.350 --> 00:11:49.320
So f of x is equal to
x to the third plus 1,

00:11:49.320 --> 00:11:51.520
so it might look
something like this.

00:11:51.520 --> 00:11:53.120
And once again,
you will be tempted

00:11:53.120 --> 00:11:54.450
to call this an odd function.

00:11:54.450 --> 00:11:58.040
But because it's shifted up, it
is no longer an odd function.

00:11:58.040 --> 00:11:59.290
You can look at that visually.

00:11:59.290 --> 00:12:03.510
So this is f of x is equal
to x to the third plus 1.

00:12:03.510 --> 00:12:05.620
If you take what's on
the right hand side

00:12:05.620 --> 00:12:07.920
and reflect it onto
to the left hand side,

00:12:07.920 --> 00:12:09.650
you would get
something like that.

00:12:09.650 --> 00:12:11.450
And then if you were
to reflect that down,

00:12:11.450 --> 00:12:13.270
you would get
something like that.

00:12:13.270 --> 00:12:16.090
So this is not an odd function.

00:12:16.090 --> 00:12:19.860
This isn't the left
reflection and then

00:12:19.860 --> 00:12:21.497
the top-bottom
reflection of what's

00:12:21.497 --> 00:12:22.830
going on on the right hand side.

00:12:22.830 --> 00:12:25.890
This over here
actually would be.


Connection between even and odd numbers and functions

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.110
In the last video on
even and odd functions,

00:00:03.110 --> 00:00:05.830
I talk about how you
shouldn't get confused

00:00:05.830 --> 00:00:09.640
between even functions and
even numbers and odd functions

00:00:09.640 --> 00:00:10.440
and odd numbers.

00:00:10.440 --> 00:00:12.900
And I said that there wasn't
any obvious connection

00:00:12.900 --> 00:00:17.490
between the word even function
and our notion of even numbers,

00:00:17.490 --> 00:00:20.740
or any connection between odd
functions and odd numbers.

00:00:20.740 --> 00:00:22.110
And, I was wrong.

00:00:22.110 --> 00:00:25.050
There actually is a
relatively obvious connection,

00:00:25.050 --> 00:00:28.780
and this was pointed out by
the YouTube user Nothias.

00:00:28.780 --> 00:00:31.500
And the connection, I
almost explicitly did it

00:00:31.500 --> 00:00:32.430
in the last example.

00:00:32.430 --> 00:00:36.170
When I showed an even function,
I showed you x squared.

00:00:36.170 --> 00:00:37.770
When I showed you
an odd function,

00:00:37.770 --> 00:00:39.490
I showed you x to
the third power.

00:00:39.490 --> 00:00:42.680
When I wanted to show
you another odd function,

00:00:42.680 --> 00:00:45.400
I showed you y is
equal to x, or f

00:00:45.400 --> 00:00:48.620
of x is equal to x
to the first power.

00:00:48.620 --> 00:00:53.160
And so you might start to notice
what Nothias pointed out, is

00:00:53.160 --> 00:00:57.460
that these archetypal
or these good examples

00:00:57.460 --> 00:01:00.000
or these simple examples
of even and odd functions,

00:01:00.000 --> 00:01:04.810
when I just have a very
simple x raised to some power,

00:01:04.810 --> 00:01:09.043
whether the power
is even or odd,

00:01:09.043 --> 00:01:12.090
it's going to tell you whether
the function is even or odd.

00:01:12.090 --> 00:01:13.860
And you want to be
very careful here.

00:01:13.860 --> 00:01:18.370
Not all even or odd functions
even have exponents in them.

00:01:18.370 --> 00:01:20.360
They could be
trigonometric functions.

00:01:20.360 --> 00:01:22.700
They might be some other
type of wacky functions.

00:01:22.700 --> 00:01:24.890
You don't have to
have exponents.

00:01:24.890 --> 00:01:27.090
It's just that these
exponents are probably

00:01:27.090 --> 00:01:30.140
where the motivations for
calling these even functions

00:01:30.140 --> 00:01:32.167
and odd functions came from.

00:01:32.167 --> 00:01:33.405
And let me just be clear.

00:01:33.405 --> 00:01:35.220
It's not just also
any polynomial--

00:01:35.220 --> 00:01:38.280
and even in the last video, when
we had x to the third plus 1,

00:01:38.280 --> 00:01:40.580
this was neither even
or odd-- but if you just

00:01:40.580 --> 00:01:45.170
have the pure x raised to some
power, then all of the sudden,

00:01:45.170 --> 00:01:49.320
the motivations for
calling them even

00:01:49.320 --> 00:01:51.470
and odd start to make
sense, because if I

00:01:51.470 --> 00:01:56.980
have f of x is equal to x
to the first power-- that's

00:01:56.980 --> 00:02:00.710
the same thing as y is
equal to x-- this is odd.

00:02:00.710 --> 00:02:05.860
And it gels with the
name because we are also

00:02:05.860 --> 00:02:08.090
raising it to an odd power.

00:02:08.090 --> 00:02:11.500
If we have f of x is
equal to x squared,

00:02:11.500 --> 00:02:14.200
we saw in the previous
video, this is even.

00:02:14.200 --> 00:02:16.240
And it gels with
the idea that we're

00:02:16.240 --> 00:02:17.979
raising it to an even power.

00:02:17.979 --> 00:02:18.770
I could keep going.

00:02:18.770 --> 00:02:21.360
If it was to the x to
the third, that is odd.

00:02:21.360 --> 00:02:22.250
I could keep going.

00:02:25.064 --> 00:02:26.960
Let me write it this way.

00:02:26.960 --> 00:02:31.560
In general, if you have f
of x is equal to x to the n,

00:02:31.560 --> 00:02:40.840
then this is odd-- odd function
if n is odd, is an odd number.

00:02:40.840 --> 00:02:48.930
And this is an even
function, if n is even.

00:02:48.930 --> 00:02:50.850
And I want to make
it very clear here.

00:02:50.850 --> 00:02:52.760
The whole point of
this video is just

00:02:52.760 --> 00:02:55.410
to clarify the motivation
for calling them

00:02:55.410 --> 00:02:57.160
even or odd functions.

00:02:57.160 --> 00:03:02.000
Not all even functions are
going to be of this form

00:03:02.000 --> 00:03:04.110
here, where it's x raised
to some even power.

00:03:04.110 --> 00:03:06.450
And not all odd functions
are going to be.

00:03:06.450 --> 00:03:08.377
And I also don't want
you to be confused

00:03:08.377 --> 00:03:10.210
that if I have something
like x to the third

00:03:10.210 --> 00:03:12.111
and then I have other
stuff past that,

00:03:12.111 --> 00:03:14.360
and you say, oh, x to the
third, that's an odd number.

00:03:14.360 --> 00:03:16.310
But this is not an odd function.

00:03:16.310 --> 00:03:19.635
Just when it's just
a pure stripped down

00:03:19.635 --> 00:03:21.769
x to the third or
x to the first can

00:03:21.769 --> 00:03:23.060
you really make that statement.

00:03:23.060 --> 00:03:26.040
But that really is probably
where the motivation comes

00:03:26.040 --> 00:03:28.600
for naming them even
or odd functions.

00:03:28.600 --> 00:03:30.770
And then the other
symmetric functions,

00:03:30.770 --> 00:03:35.820
even if they don't
involve an exponent--

00:03:35.820 --> 00:03:38.040
maybe this is some type of
trigonometric function--

00:03:38.040 --> 00:03:40.125
you're calling it
even because you're

00:03:40.125 --> 00:03:46.670
saying it has the same type
of symmetry as, say, x squared

00:03:46.670 --> 00:03:48.700
or x to an even power.

00:03:48.700 --> 00:03:51.090
So you group them all
together as even functions.

00:03:51.090 --> 00:03:53.570
And then all of these, even
though this may or may not

00:03:53.570 --> 00:03:55.730
have an exponent in it,
it has the same type

00:03:55.730 --> 00:03:59.140
of symmetries as x
raised to an odd power.

00:03:59.140 --> 00:04:01.360
So that's why call
them odd functions.

00:04:01.360 --> 00:04:08.060
Well, thank you Nothias
for pointing that out.


Recognizing direct and inverse variation

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.680
I've written some
example relationships

00:00:02.680 --> 00:00:05.170
between two variables-- in
this case between m and n,

00:00:05.170 --> 00:00:07.122
between a and b,
between x and y.

00:00:07.122 --> 00:00:08.580
And what I want to
do in this video

00:00:08.580 --> 00:00:11.700
is see if we can identify
whether the relationships are

00:00:11.700 --> 00:00:14.140
a direct relationship,
whether they vary directly,

00:00:14.140 --> 00:00:17.900
or maybe they vary inversely,
or maybe it is neither.

00:00:17.900 --> 00:00:19.450
So let's explore
it a little bit.

00:00:19.450 --> 00:00:23.024
So over here, we have
m/n is equal to 1/7.

00:00:23.024 --> 00:00:24.690
So let's see how we
can manipulate this.

00:00:24.690 --> 00:00:27.239
If we multiply both sides by n.

00:00:27.239 --> 00:00:28.280
What are we going to get?

00:00:28.280 --> 00:00:30.080
And in general, you
want to separate

00:00:30.080 --> 00:00:31.782
them so that the
two variables are

00:00:31.782 --> 00:00:33.240
on different sides
of the equation,

00:00:33.240 --> 00:00:35.320
so you can see is
it going to meet,

00:00:35.320 --> 00:00:38.040
is it going to be the pattern--
let me write it this way.

00:00:38.040 --> 00:00:40.380
m is equal to k times n.

00:00:40.380 --> 00:00:42.270
This would be direct variation.

00:00:42.270 --> 00:00:46.640
Or is it going to be the pattern
m is equal to k times 1/n?

00:00:46.640 --> 00:00:48.392
This is inverse variation.

00:00:48.392 --> 00:00:49.850
And you see in
either one of these,

00:00:49.850 --> 00:00:52.110
they're on different
sides of the equal sign.

00:00:52.110 --> 00:00:54.110
So let's take this first
relationship right now.

00:00:54.110 --> 00:00:58.480
Let's multiply both sides by n.

00:00:58.480 --> 00:01:01.210
And you get m, because
these cancel out,

00:01:01.210 --> 00:01:04.769
is equal to 1/7 times n.

00:01:04.769 --> 00:01:07.350
So this actually meets the
direct variation pattern.

00:01:07.350 --> 00:01:12.560
It's some constant times n. m is
equal to some constant times n.

00:01:12.560 --> 00:01:15.900
So this right over
here is direct.

00:01:15.900 --> 00:01:18.030
They vary directly.

00:01:18.030 --> 00:01:20.300
This is direct variation.

00:01:20.300 --> 00:01:23.315
Let's see, ab is
equal to negative 3.

00:01:23.315 --> 00:01:25.440
So if we want to separate
them-- and we could do it

00:01:25.440 --> 00:01:27.660
with either variable, we
could divide both sides.

00:01:27.660 --> 00:01:29.700
I don't know, let's
divide both sides by a.

00:01:29.700 --> 00:01:31.650
We could have done it by b.

00:01:31.650 --> 00:01:37.010
If we divide both sides by a, we
get b is equal to negative 3/a.

00:01:37.010 --> 00:01:38.910
Or we could also
write this as b is

00:01:38.910 --> 00:01:41.930
equal to negative 3 times 1/a.

00:01:41.930 --> 00:01:44.040
And once again, this is
this pattern right here.

00:01:44.040 --> 00:01:46.230
One variable is equal
to a constant times 1

00:01:46.230 --> 00:01:47.930
over the other variable.

00:01:47.930 --> 00:01:50.500
In this case, our
constant is negative 3.

00:01:50.500 --> 00:01:52.925
So over here, they
vary inversely.

00:01:56.110 --> 00:01:58.480
This is an inverse relationship.

00:01:58.480 --> 00:01:59.800
Let's try this one over here.

00:01:59.800 --> 00:02:03.590
I'll try to do it in that same
color. xy is equal to 1/10.

00:02:03.590 --> 00:02:05.920
Once again, let's try to
separate the variables,

00:02:05.920 --> 00:02:09.000
isolate them on either
side of the equation.

00:02:09.000 --> 00:02:10.647
Let's divide both sides by x.

00:02:10.647 --> 00:02:12.980
You could divide by y, because
you're really just trying

00:02:12.980 --> 00:02:16.260
to find an inverse or
direct relationship.

00:02:16.260 --> 00:02:17.700
So divide both sides by x.

00:02:17.700 --> 00:02:20.900
You get y is equal
to 1/10 over x,

00:02:20.900 --> 00:02:23.410
which is the same
thing as 1/10x, which

00:02:23.410 --> 00:02:27.460
is the same thing
as 1/10 times 1/x.

00:02:27.460 --> 00:02:31.970
So y is equal to some
constant times 1/x.

00:02:31.970 --> 00:02:39.220
Once again, this is an inverse,
y and x vary inversely.

00:02:39.220 --> 00:02:40.580
Let's do this one over here.

00:02:40.580 --> 00:02:46.260
9 times m-- I'll go to that same
orange color-- 9 times 1/m is

00:02:46.260 --> 00:02:48.260
equal to n.

00:02:48.260 --> 00:02:50.110
So this one's actually
already done for us.

00:02:50.110 --> 00:02:51.130
And it might be a
little bit clearer

00:02:51.130 --> 00:02:52.580
if we just flip this around.

00:02:52.580 --> 00:02:54.621
If we just flip the left
and the right hand side,

00:02:54.621 --> 00:02:57.530
we get n is equal
to 9 times 1/m.

00:02:57.530 --> 00:03:02.990
n is equal to some
constant times 1/m.

00:03:02.990 --> 00:03:08.750
So n varies inversely
with m, inverse.

00:03:08.750 --> 00:03:11.890
And remember, if I say
that n varies inversely

00:03:11.890 --> 00:03:16.460
with m, that also means that
m varies inversely with n.

00:03:16.460 --> 00:03:18.910
Those two things
imply each other.

00:03:18.910 --> 00:03:20.910
Now let's try it with
this expression over here.

00:03:20.910 --> 00:03:23.040
And this one's a little
bit of a trickier one,

00:03:23.040 --> 00:03:25.750
because we've already separated
the variables on both hands.

00:03:25.750 --> 00:03:28.520
And then we have this
kind of-- if this

00:03:28.520 --> 00:03:30.650
was b is equal to
1/3 times a, then

00:03:30.650 --> 00:03:32.340
we would have direct
variation, then

00:03:32.340 --> 00:03:34.570
b would vary directly with a.

00:03:34.570 --> 00:03:37.450
But in this case,
we have 1/3 minus a.

00:03:37.450 --> 00:03:41.260
And you say, hey, maybe
they're opposites, or whatever.

00:03:41.260 --> 00:03:43.860
And it actually turns
out that this is neither.

00:03:43.860 --> 00:03:45.750
This is neither.

00:03:45.750 --> 00:03:49.560
And to make that
point 100% clear,

00:03:49.560 --> 00:03:51.630
let's look at two
of these examples.

00:03:51.630 --> 00:03:53.430
In direct variation,
if you scale up

00:03:53.430 --> 00:03:55.470
one variable in
one direction, you

00:03:55.470 --> 00:03:58.190
would scale up the other
variable by the same amount.

00:03:58.190 --> 00:04:02.250
So if we have x going--
if x doubles from 1 to 2,

00:04:02.250 --> 00:04:07.000
when x is 1-- actually, I
should this with m and n.

00:04:07.000 --> 00:04:09.520
So m and n.

00:04:09.520 --> 00:04:11.880
So when-- and the way
I've written it here,

00:04:11.880 --> 00:04:14.570
although you could algebraically
manipulate it so that one looks

00:04:14.570 --> 00:04:16.060
more dependent than the other.

00:04:16.060 --> 00:04:22.310
But in this situation
where n is 1, m is 1/7.

00:04:25.360 --> 00:04:28.830
And when n is 7, m
is going to be 1.

00:04:28.830 --> 00:04:32.950
So you have the situation
that if n is scaled up by 7,

00:04:32.950 --> 00:04:36.250
then m is also scaled
up by 7, or vice versus.

00:04:36.250 --> 00:04:37.620
So it's more of a relationship.

00:04:37.620 --> 00:04:39.590
I could have expressed
n in terms of m,

00:04:39.590 --> 00:04:42.390
but when you scale
one variable up by 7,

00:04:42.390 --> 00:04:45.630
you also have to scale up
the other variable by 7.

00:04:45.630 --> 00:04:47.267
When you scale it
up by some amount,

00:04:47.267 --> 00:04:49.600
you have to scale the other
variable by the same amount.

00:04:49.600 --> 00:04:53.300
So this is direct variation.

00:04:53.300 --> 00:04:57.600
Let's take the inverse, or when
two variables vary inversely,

00:04:57.600 --> 00:04:59.870
this situation right over here.

00:04:59.870 --> 00:05:03.010
Let's take a and b.

00:05:03.010 --> 00:05:06.464
When a is equal to 1, b
is equal to negative 3.

00:05:06.464 --> 00:05:08.380
And we could do it
explicitly right over here.

00:05:08.380 --> 00:05:09.838
We can even go to
the original one.

00:05:09.838 --> 00:05:12.580
When a is equal to 1, we have
1b is equal to negative 3.

00:05:12.580 --> 00:05:14.280
b is equal to negative 3.

00:05:14.280 --> 00:05:17.950
Now when a-- if I were to take
a, and if I were to, let's say,

00:05:17.950 --> 00:05:19.480
I were to triple it.

00:05:19.480 --> 00:05:21.240
So far we're going
to multiply it by 3.

00:05:21.240 --> 00:05:23.850
So now a is 3.

00:05:23.850 --> 00:05:26.000
We have 1/3 times negative 3.

00:05:26.000 --> 00:05:27.980
So now b is negative.

00:05:27.980 --> 00:05:30.360
Notice we didn't
multiply b times 3 here.

00:05:30.360 --> 00:05:36.380
Now we divided b times 3, or we
divided b by 3, I should say.

00:05:36.380 --> 00:05:39.750
Or another way is we
multiplied by 1/3.

00:05:39.750 --> 00:05:44.490
So if you scale up a by 3,
you're scaling down b by 3.

00:05:44.490 --> 00:05:46.560
So they're varying inversely.

00:05:46.560 --> 00:05:48.250
What you're going to
see in this neither

00:05:48.250 --> 00:05:50.333
is that neither of these
are going to be the case.

00:05:50.333 --> 00:05:53.100
So let's try it out.

00:05:53.100 --> 00:05:58.090
I'll do it in that same
green color, that same green.

00:05:58.090 --> 00:06:00.010
So we have a and b.

00:06:00.010 --> 00:06:04.910
So when a is-- I don't know,
what a is 1, what is b?

00:06:04.910 --> 00:06:11.310
1/3 minus 1, that's 1/3 minus
3/3, that's negative 2/3.

00:06:11.310 --> 00:06:14.360
And then let's
divide just for fun.

00:06:14.360 --> 00:06:18.220
Let's divide a by 3.

00:06:18.220 --> 00:06:21.620
So if a goes to 1/3, so over
here we're dividing by 3,

00:06:21.620 --> 00:06:24.490
or you could say we're
multiplying by 1/3.

00:06:24.490 --> 00:06:29.260
So if a is 1/3, then b
is equal to 0, right?

00:06:29.260 --> 00:06:31.880
If a is 1/3, b is equal to 0.

00:06:31.880 --> 00:06:34.420
So notice, if this
was direct variation,

00:06:34.420 --> 00:06:37.580
we would be multiplying this by
1/3 as well-- which, clearly,

00:06:37.580 --> 00:06:38.490
we didn't.

00:06:38.490 --> 00:06:40.350
And if this was
inverse variation,

00:06:40.350 --> 00:06:43.120
if they varied inversely, we
would be multiplying by 3,

00:06:43.120 --> 00:06:44.180
which clearly we didn't.

00:06:44.180 --> 00:06:46.640
We just got some other number,
which actually ended up

00:06:46.640 --> 00:06:48.310
the scaling actually
didn't matter.

00:06:48.310 --> 00:06:50.100
What happens is
that things really

00:06:50.100 --> 00:06:52.560
just got shifted by some amount.

00:06:52.560 --> 00:06:54.736
They got shifted by 2/3.

00:06:58.250 --> 00:07:02.300
These neither vary
directly nor inversely,

00:07:02.300 --> 00:07:04.920
this last one right over here.


Direct and inverse variation

WEBVTT
Kind: captions
Language: en

00:00:00.385 --> 00:00:01.760
I want to talk a
little bit about

00:00:01.760 --> 00:00:04.280
direct and inverse variations.

00:00:04.280 --> 00:00:07.150
So I'll do direct variation
on the left over here.

00:00:07.150 --> 00:00:10.930
And I'll do inverse variation,
or two variables that

00:00:10.930 --> 00:00:15.590
vary inversely, on the
right-hand side over here.

00:00:15.590 --> 00:00:18.690
So a very simple definition
for two variables

00:00:18.690 --> 00:00:21.660
that vary directly would
be something like this.

00:00:21.660 --> 00:00:27.010
y varies directly with x if y is
equal to some constant with x.

00:00:27.010 --> 00:00:29.450
So we could rewrite
this in kind of English

00:00:29.450 --> 00:00:39.020
as y varies directly with x.

00:00:39.020 --> 00:00:41.720
And if this constant
seems strange to you,

00:00:41.720 --> 00:00:44.710
just remember this could be
literally any constant number.

00:00:44.710 --> 00:00:49.500
So let me give you a bunch
of particular examples

00:00:49.500 --> 00:00:52.300
of y varying directly with x.

00:00:52.300 --> 00:00:55.470
You could have y is equal to x.

00:00:55.470 --> 00:00:57.430
Because in this situation,
the constant is 1.

00:00:57.430 --> 00:00:58.430
We didn't even write it.

00:00:58.430 --> 00:01:02.020
We could write y is
equal to 1x, then k is 1.

00:01:02.020 --> 00:01:04.660
We could write y is equal to 2x.

00:01:04.660 --> 00:01:07.850
We could write y
is equal to 1/2 x.

00:01:07.850 --> 00:01:11.970
We could write y is
equal to negative 2x.

00:01:11.970 --> 00:01:14.800
We are still varying directly.

00:01:14.800 --> 00:01:18.260
We could have y is
equal to negative 1/2 x.

00:01:18.260 --> 00:01:22.800
We could have y is
equal to pi times x.

00:01:22.800 --> 00:01:25.987
We could have y is equal
to negative pi times x.

00:01:25.987 --> 00:01:27.570
I don't want to beat
a dead horse now.

00:01:27.570 --> 00:01:28.840
I think you get the point.

00:01:28.840 --> 00:01:32.190
Any constant times x--
we are varying directly.

00:01:32.190 --> 00:01:35.380
And to understand this maybe
a little bit more tangibly,

00:01:35.380 --> 00:01:36.790
let's think about what happens.

00:01:36.790 --> 00:01:39.110
And let's pick one
of these scenarios.

00:01:39.110 --> 00:01:41.680
Well, I'll take a positive
version and a negative version,

00:01:41.680 --> 00:01:43.950
just because it might not
be completely intuitive.

00:01:43.950 --> 00:01:46.720
So let's take the version
of y is equal to 2x,

00:01:46.720 --> 00:01:50.500
and let's explore why we
say they vary directly

00:01:50.500 --> 00:01:51.990
with each other.

00:01:51.990 --> 00:01:54.020
So let's pick a
couple of values for x

00:01:54.020 --> 00:01:57.880
and see what the resulting
y value would have to be.

00:01:57.880 --> 00:02:03.410
So if x is equal to 1, then
y is 2 times 1, or is 2.

00:02:03.410 --> 00:02:07.470
If x is equal to 2,
then y is 2 times

00:02:07.470 --> 00:02:10.630
2, which is going
to be equal to 4.

00:02:10.630 --> 00:02:13.900
So when we doubled x,
when we went from 1 to 2--

00:02:13.900 --> 00:02:17.800
so we doubled x-- the
same thing happened to y.

00:02:17.800 --> 00:02:21.160
We doubled y.

00:02:21.160 --> 00:02:24.320
So that's what it means when
something varies directly.

00:02:24.320 --> 00:02:26.410
If we scale x up by
a certain amount,

00:02:26.410 --> 00:02:29.260
we're going to scale up
y by the same amount.

00:02:29.260 --> 00:02:31.770
If we scale down
x by some amount,

00:02:31.770 --> 00:02:35.010
we would scale down
y by the same amount.

00:02:35.010 --> 00:02:37.010
And just to show you it
works with all of these,

00:02:37.010 --> 00:02:42.010
let's try the situation with
y is equal to negative 2x.

00:02:42.010 --> 00:02:43.630
I'll do it in magenta.

00:02:43.630 --> 00:02:46.272
y is equal to
negative-- well, let

00:02:46.272 --> 00:02:48.480
me do a new example that I
haven't even written here.

00:02:48.480 --> 00:02:52.580
Let's try y is equal
to negative 3x.

00:02:52.580 --> 00:02:57.220
So once again, let
me do my x and my y.

00:02:57.220 --> 00:03:00.170
When x is equal to 1, y is
equal to negative 3 times

00:03:00.170 --> 00:03:01.910
1, which is negative 3.

00:03:01.910 --> 00:03:06.880
When x is equal to 2,
so negative 3 times

00:03:06.880 --> 00:03:08.950
2 is negative 6.

00:03:08.950 --> 00:03:11.840
So notice, we multiplied.

00:03:11.840 --> 00:03:16.030
So if we scaled-- let me do
that in that same green color.

00:03:16.030 --> 00:03:19.220
If we scale up x by 2-- it's
a different green color,

00:03:19.220 --> 00:03:22.845
but it serves the purpose--
we're also scaling up y by 2.

00:03:22.845 --> 00:03:25.230
To go from 1 to 2,
you multiply it by 2.

00:03:25.230 --> 00:03:27.630
To go from negative
3 to negative 6,

00:03:27.630 --> 00:03:29.180
you're also multiplying by 2.

00:03:29.180 --> 00:03:32.680
So we grew by the
same scaling factor.

00:03:32.680 --> 00:03:37.760
And if you wanted to go
the other way-- let's try,

00:03:37.760 --> 00:03:42.460
I don't know, let's
go to x is 1/3.

00:03:42.460 --> 00:03:46.220
If x is 1/3, then y is going
to be-- negative 3 times

00:03:46.220 --> 00:03:48.280
1/3 is negative 1.

00:03:48.280 --> 00:03:52.520
So notice, to go from 1
to 1/3, we divide by 3.

00:03:52.520 --> 00:03:54.840
To go from negative
3 to negative 1,

00:03:54.840 --> 00:03:56.660
we also divide by 3.

00:03:56.660 --> 00:04:00.080
We also scale down
by a factor of 3.

00:04:00.080 --> 00:04:02.490
So whatever direction
you scale x in,

00:04:02.490 --> 00:04:05.190
you're going to have the
same scaling direction as y.

00:04:05.190 --> 00:04:08.680
That's what it means
to vary directly.

00:04:08.680 --> 00:04:11.000
Now, it's not always so clear.

00:04:11.000 --> 00:04:14.640
Sometimes it will be obfuscated.

00:04:14.640 --> 00:04:16.970
So let's take this
example right over here.

00:04:16.970 --> 00:04:18.672
y is equal to negative 3x.

00:04:18.672 --> 00:04:20.880
And I'm saving this real
estate for inverse variation

00:04:20.880 --> 00:04:21.730
in a second.

00:04:21.730 --> 00:04:24.150
You could write it like this,
or you could algebraically

00:04:24.150 --> 00:04:25.100
manipulate it.

00:04:25.100 --> 00:04:28.630
You could maybe divide both
sides of this equation by x,

00:04:28.630 --> 00:04:34.080
and then you would get y/x
is equal to negative 3.

00:04:34.080 --> 00:04:36.810
Or maybe you divide
both sides by x,

00:04:36.810 --> 00:04:39.070
and then you divide
both sides by y.

00:04:39.070 --> 00:04:42.590
So from this, so if you
divide both sides by y now,

00:04:42.590 --> 00:04:48.530
you could get 1/x is equal
to negative 3 times 1/y.

00:04:48.530 --> 00:04:50.930
These three statements,
these three equations,

00:04:50.930 --> 00:04:52.890
are all saying the same thing.

00:04:52.890 --> 00:04:54.330
So sometimes the
direct variation

00:04:54.330 --> 00:04:55.780
isn't quite in your face.

00:04:55.780 --> 00:04:59.080
But if you do this, what I did
right here with any of these,

00:04:59.080 --> 00:05:01.500
you will get the
exact same result.

00:05:01.500 --> 00:05:03.690
Or you could just try
to manipulate it back

00:05:03.690 --> 00:05:04.817
to this form over here.

00:05:04.817 --> 00:05:06.400
And there's other
ways we could do it.

00:05:06.400 --> 00:05:09.650
We could divide both sides of
this equation by negative 3.

00:05:09.650 --> 00:05:14.520
And then you would get
negative 1/3 y is equal to x.

00:05:14.520 --> 00:05:16.560
And now, this is kind
of an interesting case

00:05:16.560 --> 00:05:21.380
here because here, this is
x varies directly with y.

00:05:21.380 --> 00:05:24.674
Or we could say x is
equal to some k times y.

00:05:24.674 --> 00:05:25.840
And in general, that's true.

00:05:25.840 --> 00:05:28.080
If y varies directly
with x, then

00:05:28.080 --> 00:05:34.456
we can also say that x
varies directly with y.

00:05:34.456 --> 00:05:36.080
It's not going to be
the same constant.

00:05:36.080 --> 00:05:38.496
It's going to be essentially
the inverse of that constant,

00:05:38.496 --> 00:05:40.595
but they're still
directly varying.

00:05:40.595 --> 00:05:41.970
Now with that
said, so much said,

00:05:41.970 --> 00:05:43.770
about direct variation,
let's explore

00:05:43.770 --> 00:05:45.870
inverse variation a little bit.

00:05:45.870 --> 00:05:48.622
Inverse variation--
the general form,

00:05:48.622 --> 00:05:49.830
if we use the same variables.

00:05:49.830 --> 00:05:51.538
And it always doesn't
have to be y and x.

00:05:51.538 --> 00:05:52.620
It could be an a and a b.

00:05:52.620 --> 00:05:55.830
It could be a m and an n.

00:05:55.830 --> 00:05:58.430
If I said m varies
directly with n,

00:05:58.430 --> 00:06:01.510
we would say m is equal
to some constant times n.

00:06:01.510 --> 00:06:03.279
Now let's do inverse variation.

00:06:03.279 --> 00:06:04.820
So if I did it with
y's and x's, this

00:06:04.820 --> 00:06:09.740
would be y is equal to
some constant times 1/x.

00:06:09.740 --> 00:06:12.070
So instead of being
some constant times x,

00:06:12.070 --> 00:06:14.215
it's some constant times 1/x.

00:06:14.215 --> 00:06:15.840
So let me draw you
a bunch of examples.

00:06:15.840 --> 00:06:18.670
It could be y is equal to 1/x.

00:06:18.670 --> 00:06:23.380
It could be y is equal to 2
times 1/x, which is clearly

00:06:23.380 --> 00:06:25.810
the same thing as 2/x.

00:06:25.810 --> 00:06:31.370
It could be y is equal
to 1/3 times 1/x, which

00:06:31.370 --> 00:06:34.190
is the same thing as 1 over 3x.

00:06:34.190 --> 00:06:39.140
it could be y is equal
to negative 2 over x.

00:06:39.140 --> 00:06:42.200
And let's explore this, the
inverse variation, the same way

00:06:42.200 --> 00:06:44.590
that we explored the
direct variation.

00:06:44.590 --> 00:06:47.520
So let's pick-- I don't know/
let's pick y is equal to 2/x.

00:06:50.885 --> 00:06:54.360
And let me do that
same table over here.

00:06:54.360 --> 00:06:55.500
So I have my table.

00:06:55.500 --> 00:06:57.830
I have my x values
and my y values.

00:06:57.830 --> 00:07:01.660
If x is 1, then y is 2.

00:07:01.660 --> 00:07:06.300
If x is 2, then 2
divided by 2 is 1.

00:07:06.300 --> 00:07:09.370
So if you multiply x
by 2, if you scale it

00:07:09.370 --> 00:07:12.670
up by a factor of 2,
what happens to y?

00:07:12.670 --> 00:07:15.000
y gets scaled down
by a factor of 2.

00:07:15.000 --> 00:07:16.830
You're dividing by 2 now.

00:07:16.830 --> 00:07:17.970
Notice the difference.

00:07:17.970 --> 00:07:22.580
Here, however we scaled x, we
scaled up y by the same amount.

00:07:22.580 --> 00:07:25.140
Now, if we scale up
x by a factor, when

00:07:25.140 --> 00:07:30.020
we have inverse variation, we're
scaling down y by that same.

00:07:30.020 --> 00:07:32.550
So that's where the
inverse is coming from.

00:07:32.550 --> 00:07:33.800
And we could go the other way.

00:07:33.800 --> 00:07:36.120
If we made x is equal to 1/2.

00:07:36.120 --> 00:07:40.420
So if we were to
scale down x, we're

00:07:40.420 --> 00:07:42.340
going to see that it's
going to scale up y.

00:07:42.340 --> 00:07:45.050
Because 2 divided by 1/2 is 4.

00:07:45.050 --> 00:07:46.819
So here we are scaling up y.

00:07:46.819 --> 00:07:48.610
So they're going to do
the opposite things.

00:07:48.610 --> 00:07:51.130
They vary inversely.

00:07:51.130 --> 00:07:53.650
And you could try it with
the negative version of it,

00:07:53.650 --> 00:07:54.170
as well.

00:07:54.170 --> 00:07:57.500
So here we're multiplying by 2.

00:07:57.500 --> 00:08:00.310
And once again, it's not
always neatly written for you

00:08:00.310 --> 00:08:01.230
like this.

00:08:01.230 --> 00:08:04.170
It can be rearranged in a
bunch of different ways.

00:08:04.170 --> 00:08:07.680
But it will still
be inverse variation

00:08:07.680 --> 00:08:09.722
as long as they're
algebraically equivalent.

00:08:09.722 --> 00:08:11.930
So you can multiply both
sides of this equation right

00:08:11.930 --> 00:08:13.720
here by x.

00:08:13.720 --> 00:08:17.340
And you would get
xy is equal to 2.

00:08:17.340 --> 00:08:19.130
This is also inverse variation.

00:08:19.130 --> 00:08:21.440
You would get this exact
same table over here.

00:08:21.440 --> 00:08:24.420
You could divide both sides
of this equation by y.

00:08:24.420 --> 00:08:31.330
And you could get x is
equal to 2/y, which is also

00:08:31.330 --> 00:08:33.870
the same thing as 2 times 1/y.

00:08:33.870 --> 00:08:37.141
So notice, y varies
inversely with x.

00:08:37.141 --> 00:08:39.140
And you could just
manipulate this algebraically

00:08:39.140 --> 00:08:43.320
to show that x varies
inversely with y.

00:08:43.320 --> 00:08:50.200
So y varies inversely with x.

00:08:50.200 --> 00:08:52.639
This is the same thing as
saying-- and we just showed it

00:08:52.639 --> 00:08:54.180
over here with a
particular example--

00:08:54.180 --> 00:09:00.239
that x varies inversely with y.

00:09:00.239 --> 00:09:01.280
And there's other things.

00:09:01.280 --> 00:09:04.270
We could take this and
divide both sides by 2.

00:09:04.270 --> 00:09:08.000
And you would get
y/2 is equal to 1/x.

00:09:08.000 --> 00:09:09.720
There's all sorts
of crazy things.

00:09:09.720 --> 00:09:12.332
And so in general, if you
see an expression that

00:09:12.332 --> 00:09:13.790
relates to variables,
and they say,

00:09:13.790 --> 00:09:17.970
do they vary inversely or
directly or maybe neither?

00:09:17.970 --> 00:09:20.160
You could either try to
do a table like this.

00:09:20.160 --> 00:09:22.550
If you scale up x
by a certain amount

00:09:22.550 --> 00:09:24.970
and y gets scaled up by
the same amount, then

00:09:24.970 --> 00:09:26.570
it's direct variation.

00:09:26.570 --> 00:09:29.487
If you scale up x
by some-- and you

00:09:29.487 --> 00:09:31.320
might want to try a
couple different times--

00:09:31.320 --> 00:09:34.180
and you scale down y, you
do the opposite with y,

00:09:34.180 --> 00:09:35.780
then it's probably
inverse variation.

00:09:39.130 --> 00:09:41.510
A surefire way of knowing
what you're dealing with

00:09:41.510 --> 00:09:44.320
is to actually algebraically
manipulate the equation

00:09:44.320 --> 00:09:47.150
so it gets back to either this
form, which would tell you

00:09:47.150 --> 00:09:50.480
that it's inverse variation, or
this form, which would tell you

00:09:50.480 --> 00:09:52.676
that it is direct variation.