Once again we're studying domain and range in class and I encountered this problem.
If $f(x)$ and $g(x)$ are both invertible functions, and the domain and range of each
function is the set of real numbers, express $\bigl(f\circ g\circ (f^{-1})\bigr)^{-1}(x)$ as a composition of the functions $f(x)$, $g(x)$, and their inverses.
Any help with this problem would be greatly appreciated.
Best Answer
Forget the $x$. You can solve this at the function level.
They key identity is
$$ ( f \circ g )^{-1} = g^{-1} \circ f^{-1}$$
... which you can extend to a chain of compositions.
And, of course
$$ (f^{-1})^{-1} = f$$
Just apply these to multiply out and simplify the composite inversions.