If I have two functions defined from $\mathbb{R^3} \to \mathbb{R}$, can they be inverses?

Question

I am taking an economics class and I am not getting some straight answers about the conditions under which I can say that a map from $$\Bbb{R}^3 \rightarrow \Bbb{R}$$ can be reduced to a map from $$\Bbb{R} \rightarrow \Bbb{R}$$

Specifically, I have a function $$v(p_x,p_y,m)$$ and a function $$e(p_x,p_y,u)$$

and my professor says (I've phrased it mathematically) assuming $p_x$ and $p_y$ are constant then

$$(e \circ v)(m) = m$$ $$(v \circ e)(u) = u$$

In what sense can I call these inverses? Is it necessary to have $p_x$ and $p_y$ fixed for $e$ and $v$ to be inverses?

Answer 1

If $p_x, p_y$ are each fixed, then what you really have is $v:\mathbb{R}\to \mathbb{R}$, and $e$ takes you back. They are then inverses provided they satisfy the two conditions you gave. More specifically, one could call it $v_{p_x,p_y}(m)$, since the function depends on which specific fixed values of $p_x, p_y$ you have. Hence, using this notation, $v_{p_x,p_y}$ and $e_{p_x,p_y}(m)$ are inverses. However this notation is not mandatory, just helpful for understanding.

If $p_x, p_y$ are not fixed, then $v$ and $e$ cannot be inverses, because the domain of $v$ is $\mathbb{R}^3$, while the codomain of $e$ is $\mathbb{R}$. For functions $f,g$ to be inverses, you need (to start) that $f:U\to V$, and $g:V\to U$.

Answer 2

An example might help: Consider the familiar equation $y=mx+b$. You could think of this as a function $f:\mathbb R^3 \to \mathbb R$ and write $y=f(x,m,b)$. But in most cases we take $m$ and $b$ as fixed parameters -- the slope and the $y$-intercept of a line -- and in that usual case we think of this as a function $g:\mathbb R \to \mathbb R$ and write $y=g(x)$. One could of course emphasize that $g$ depends on the parameters $m$ and $b$ by writing $y=g_{m,b}(x)$, but usually we omit that because context makes it clear what we mean.