Question

Let f : X → Y be an invertible function. Show that the inverse of f¹ is f, i.e., that (f ^-1)^-1 = f

Answer 1

Let f : X → Y is invertible 

⇒ f is one and onto and 

f^-1 : Y ⇒ X is defined as f^-1(y) = x 

y : f (x) ∀ x ∈ X and y ∈ Y 

let y₁,y₂∈ y f^-1 (y₁) = f^-2(y₂) 

fof¹ (y₁) = fof¹ (y2) 

I_y (y₁) = I_y(y₂) 

⇒ y₁ = y₂∴ f₁ is one-one 

∀ x ∈ X, ∋ y ∈ Y such that 

f¹(y) = x, hence f¹ is onto 

hence invertible. 

let g = (f¹)^-1 

gof^-1 = I_y and f^-1og = lx 

∀ x ∈ X, I_x (x) = x 

fof^-1(x) = f^-1 [g(x)] = x 

fof^-1 [g (x)] = f (x) 

(fof^-1) (g (x)) = f (x) 

g(x) = f (x) 

g = f 

(f^-1)^-1 = f