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# 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

(g_{1} and Then, for all y ∈ Y, g_{2})

fog_{1} (y) = I_{Y} (y) = fog_{2} (y)

⇒ f (g_{1} (y)) = f (g_{2} (y))

⇒ g_{1} (y) = g_{2} (y)

[f is invertible ⇒ f is one-one]

⇒ g_{1} = g_{2}

[g is one-one]

Hence, f has unique inverse

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, fog_{1} (y) = I_{Y} (y) = fog_{2} (y), f is invertible ⇒ f is one-one hence f has unique inverse

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