Let f :X $\to$ Y be an invertible function. Show that f has unique inverse. (Hint: suppose g 1 and g 2 are two inverses of f. Then for all y∈ Y, fog 1 (y) = I Y (y) = fog 2 (y). Use one-one ness of f).

Let f : X → Y be an invertible function. Also, suppose f has two inverses (say ). Then, for all y ∈Y, we have: 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 a unique inverse.

Question:

Let f :X$\to$ 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) = I_Y(y) = fog₂(y). Use one-one ness of f).

CBSE CLASS XII

Updated On: Aug 24, 2023

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Solution and Explanation

Let f : X → Y be an invertible function. Also, suppose f has two inverses (say ).
Then, for all y ∈Y, we have: 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].