Find inverse composition I've been stuck for an hour trying to solve this composition function:
If $f(\sqrt[7] x)=g(3x-5)$, then $[(g^{-1}\circ f)^{-1} \circ g^{-1}](x)=\ldots$
My friend gave me a hint that I should find $f^{-1}$ first, but I don't really get it. Can someone help me to understand this problem?

We are given $f(x^{\frac{1}{7}})=g(3x-5)$ and asked to find ${(g^{-1}\circ f)}^{-1}\circ g^{-1}$.
Since you've been asked to find this, you can assume $g$ and $(g^{-1}\circ f)$ are invertible. Therefore $g \circ(g^{-1}\circ f)=f$ is invertible.
Now,
$$f(x^{\frac{1}{7}})=g(3x-5)\iff f(x)=g(3x^7-5)\iff f^{-1}(x)={\left(\frac{g^{-1}(x)+5}{3}\right)}^{1\over7}$$
These are equivalent statements, so you've essentially been given the last statement, and asked to find ${(g^{-1}\circ f)}^{-1}\circ g^{-1}=(f^{-1}\circ g)\circ g^{-1}=f^{-1}$.
Therefore you can choose any invertible $g(x)$, and this is as far as you can get.
