# Let f: X → Y be an invertible function.

Question:

Let $f: X \rightarrow Y$ be an invertible function. Show that the inverse of $f^{-1}$ is $f$, i.e., $\left(f^{-1}\right)^{-1}=f .$

Solution:

Let $f: X \rightarrow Y$ be an invertible function.

Then, there exists a function gY → X such that gof = IXand fo= IY.

Here, $f^{-1}=g$

Now, gof $=\left.\right|_{x}$ and fog $=\left.\right|_{y}$

$\Rightarrow f^{-1}$ of $=\left.\right|_{X}$ and $f \circ f^{-1}=\left.\right|_{Y}$

Hence, $f^{-1}: Y \rightarrow X$ is invertible and $f$ is the inverse of $f^{-1}$

i.e., $\left(f^{-1}\right)^{-1}=f$