Question

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

Answer 1

f:X → Y is an inversible function. then g: Y → X exists in such a way that
$gof=I_x\:and\:fog=I_y$
therefore , $f^{-1}=g$ [e.g., g is inverse of f ]
now, $gof=I_x\:\implies f^{-1}of=I_x$
similarly, $fog=I_x\:\implies fof^{-1}=I_y$
Thus, f⁻¹: Y→X is invertible and f is the inverse of $f^{-1}$.
Therefore, (f⁻¹)⁻¹ = f.
[hence proved]

Answer 2

Answer:

f:X → Y is an inversible function. then g: Y → X exists in such a way that

therefore ,  [e.g., g is inverse of f ]

now,

similarly,

Thus, f⁻¹: Y→X is invertible and f is the inverse of .

Therefore, (f⁻¹)⁻¹ = f.

[hence proved]

Step-by-step explanation: