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# Let f : X → Y be an invertible function. Show that the inverse of f^{ –1} is f, i.e., (f ^{–1} )^{ –1} = f.

Q. 12 Let $f : X \rightarrow Y$ be an invertible function. Show that the inverse of $f^{-1}$ is $f$, i.e.,
$(f^{-1})^{-1} = f$

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$f : X \rightarrow Y$

To prove: $(f^{-1})^{-1} = f$

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

Then there is    $g:Y\rightarrow X$  such that   $gof =I_x$  and $fog=I_y$

Also,       $f^{-1}= g$

$gof =I_x$  and $fog=I_y$

$\Rightarrow$        $f^{-1}of = I_x$            and         $fof^{-1} = I_y$

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

i.e. $(f^{-1})^{-1} = f$

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