Let f: X → 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 → Y be an invertible function.

Then, there exists a function g: Y → X such that gof = I_Xand fog = I_Y.

Here, f⁻¹ = g.

Now, gof = I_Xand fog = I_Y

⇒ f⁻¹of = I_Xand fof⁻¹= I_Y

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

i.e., (f⁻¹)⁻¹ = f.

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