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 g: Y → X such that gof = IXand fog = 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., $\left(f^{-1}\right)^{-1}=f$
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