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 invertible function.

∴ f is one-one and onto ⇒g : Y → X, where g is also one-one and onto such that

gof(x) = Ix and fog(y) = Iy ⇒ g = f-1.

Now, f-1o(f-1)-1 = I

and fo[f-1o(f-1)-1] = foI

or (fof-1)o(f-1)-1 = f

Io(f-1)-1 = f

⇒ (f-1)-1 = f