What does $f(x,x)$ mean?

Question

In the context of partial derivatives $\partial_x f(x,x)$ are usually defined to mean $\partial_1 f(x,x)$, as a way to point out that you differentiate with respect to the first variable, not the variable $x$ ($\color{red}{?}$). However, what does $f(x,x)$ even mean? Can a two-variabled function really be dependent of one variable? or is it a shorthand way of writing $f(x\mapsto x,y\mapsto x)$?

Answer 1

It's just function composition as I think you noted. For example, if $$ f(x,y)=x^2+y^3 $$ then $$ f(x,x)=x^2+x^3 $$

In response to your last comment, "Why would one want to reduce a function to one variable, and still call it a two-variabled function?"

You may care about a function's behavior along some curve, in this case $y=x$. This comes up in max/minimization with inequality constraints while checking the function along the boundary for extrema.

Answer 2

This is one of those areas where you have to predict what the author is trying to say to work it out, but ideally if $f(,)$ is defined as $f(x,~y)=\dots$ then

$$\frac{\partial f(x,~ x)}{\partial x} = \bigg({\rm D}_1~f\bigg)(x,~x)$$

But it is possible the author is misusing notation and meaning

$$\frac{\partial f(x,~ x)}{\partial x} = \frac{\rm d}{{\rm d} x}\bigg( f(x, x) \bigg)$$

You'll probably have to post a link to the text or just figure out which one makes more sense to work out which is meant.