Let f : X → Y be an invertible function. Show that f has unique inverse. (Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y, fog1 (y) = 1Y (y) = fog2 (y). Use one-one ness of f)

Solution:

A function is a process or a relation that associates each element 'a' of a non-empty set A, to a single element 'b' of another non-empty set B.

According to the given problem,

Let f : X → Y be an invertible function.

Also, suppose f has two inverses

(g₁ and Then, for all y ∈ Y, g₂)

fog₁ (y) = I_Y (y) = fog₂ (y)

⇒ f (g₁ (y)) = f (g₂ (y))

⇒ g₁ (y) = g₂ (y)

[f is invertible ⇒ f is one-one]

⇒ g₁ = g₂

[g is one-one]

Hence, f has unique inverse

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.3 Question 10

Let f : X → Y be an invertible function. Show that f has unique inverse. (Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y, fog1 (y) = 1Y (y) = fog2 (y). Use one-one ness of f)

Summary:

For the function f : X → Y be an invertible function,  fog₁ (y) = I_Y (y) = fog₂ (y), f is invertible ⇒ f is one-one hence f has unique inverse