Question

Let f : X → Y be an invertible function. Show that f has a unique inverse.

Answer 1

If f is invertible, then gof(x) = Ix and fog(y) = Iy.

⇒ f is one-one and onto.

Let there be two inverses g1 and g2

∴ fog1 (y)= Iy, f0g2(y) = Iy

Iy being unique for a given function f, we get

g1(y) = g2(y) [ ∵ f is one-one and onto.]

∴ f has a unique inverse.