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

Answers (1)

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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