Finding demand functions for an unusual utility function I have a utility function: $U = x + \min\{x,y\}$

I want to draw the indifference curve and find the demand functions. Will it be the case of the usual perfect complements? 

Also, what preferences could such a utility function represent?

Will the optimal solution be at the kink?
I assume you know how does $\min\{x,y\}$ look like? In order to draw utility function of interest, you need to consider cases: $u(x,y)=x+\min\{x,y\}=\begin{cases}2x, \;\; \mathrm{for} \;\; x \leq y \\ x+y, \;\; \mathrm{for} \;\; x &gt; y\end{cases}$ 

With $x$ on horizontal and $y$ on vertical axis:



Not sure about the "usual" perfect complements. It is more like a combination of substitutes (below $y=x$) and complements (above $y=x$).

Also, take a look here: Identifying utility function and Algebraic approach towards convexity where you can see more graphs. 

EDIT.

As for the demand function, Finding demand function given a utility min(x,y) function.