Let f : X → Y be an invertible function. Show that the inverse of f1

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

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

Answer»

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