Composition of Invertible Functions

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.

• Tried drawing a diagram? One of those sloppy ones with big circles for sets and dots for elements. Pretend you have two copies of $\mathbb R$ and $f$ is $\mathbb R_1\to\mathbb R_2$ whereas $g$ is $\mathbb R_1\to\mathbb R_1$. – Henning Makholm Dec 22 '14 at 22:30

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.

• Associativity of composition deserves a mention though! – GPerez Dec 22 '14 at 23:33