Composition of an inverse function with another function

Question

Suppose I have two functions $f(x)$ and $g(y)$. Then what is $f^{-1}$ composed with $g$, i.e.:

$f^{-1}\circ g$ ?

To me it looks like the value of $x$ when $f(x) = g(y)$, but I am not entirely sure. Could someone shed more light on this please.

Answer 1

I think things will be clearer if you don’t talk about $x$ and $y$. Both $f$ and $g$ are functions, and under certain conditions, $f$’s inverse $f^{-1}$ is defined.

Now, if you start with a number $z$ and want to know what happens to it under $f^{-1}\circ g$, you first apply $g$ to get $w=g(z)$. This $w$, if you’re lucky, happens to be equal to $f(u)$ for $u$ in the domain of $f$. That is, $w=g(z)=f(u)$. Then $\bigl(f^{-1}\circ g\bigr)(z)=u$.

Try it with $g=\sin$ and $f=$ the cubing function. You should get a nice, always-defined formula for $f^{-1}\circ g$ .