can 食べよう mean

  1. lets eat / shall (we) eat (suggesting)
  2. should eat (expressing an opinion)

?

eg;

  1. 昼食を食べよう lets have lunch
  2. お昼は、どこで食べようか? where _should_ we eat for lunch?

Which is correct meaning for ~よう/こう/おう/ろう....(verb volitional form)
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question
grammar
can 食べよう mean; 1. lets eat / shall (we) eat 2. should eat?
You cannot expect a one-to-one correspondence for such basic function words.
The volitional form can be translated into English as _will_ , _let's_ ,
_shall_ , _should_ and so on depending on the context. And _should_ can be
translated into Japanese as (よ)う, べき, はず, (れ)ばいい and so on depending on the
intended meaning.

  * 昼食を食べよう。  
Let's have lunch.

  * お昼は、どこで食べようか?  
Where should we eat for lunch?

In English, _should_ in a question can express someone's volition or casual
suggestion, so these translations are both fine. But when you say something
like "You should stop smoking", this _should_ is stronger, and you usually
need something stronger than the (よ)う-form.
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answer
If you are a beginner, you should learn the following three usages.

**しよう (with a flat or falling intonation)**

This pattern is used when you suggest that you and the listener do something
together. It is often translated as " _Let's ..._ "

> 昼食を食べよう。  
>  Let’s have lunch.

**しようか with a falling intonation**

This pattern is used when you invite the listener to decide something with you
about something you are going to do together. It is often used with a question
word (interrogative) such as なに, どこ, and いつ.

> お昼は、どこで食べようか。(↓)  
>  Where are we going to eat for lunch?

> 予約をしようか。(↓)  
>  Why don’t we make a reservation?

**しようか with a rising intonation**

This pattern is used when you offer to do something for the listener.

> 予約をしようか。(↑)  
>  Shall I make a reservation for you?
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answer