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

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Solution

Let $f :\to Y$ be on invertibble function:
Then three exists a function $g : Y \to X$ such that $g o f = I_{x}$ and $f o g = I_{y}$
here $f^{- 1} = g$
now $g o f = I_{x}$ and $f o g = I_{y}$
$f^{- 1}$ of $I_{x}$ and $f o f^{- 1} = I_{y}$
hence $f^{- 1} : Y \to X$ is invertible and $f$ is the inverse of $f^{- 1}$ i.e., ${(f^{- 1})}^{- 1} = f .$

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