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 $f : X \to Y$ be an invertible function.
Then, there exists a function $g : Y \to X$ such that gof $= I_{X}$ and fog $= I_{Y} .$
Here, $f^{- 1} = g .$
Now, gof $= I_{x}$ and fog $= I_{Y}$
Therefore, $f^{- 1}$ 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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