Share

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) = Iy(Y) = Fog2(Y). Use One-one Ness of F). - CBSE (Commerce) Class 12 - Mathematics

ConceptComposition of Functions and Invertible Function

Question

Let fX → 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 ∈ Yfog1(y) = IY(y) = fog2(y). Use one-one ness of f).

Solution

Let fX → Y be an invertible function.

Also, suppose f has two inverses (say g_1 and g_2)).

Then, for all y ∈Y, we have:

fog_1(y) = I_y (y) = fog_2 (y)

=> f(g_1(y)) = f(g_2(y))             [f is invertible => f is one-one]

=> g_1 = g_2      [g is one- one]

Hence, f has a unique inverse.

Is there an error in this question or solution?

APPEARS IN

NCERT Solution for Mathematics Textbook for Class 12 (2018 to Current)
Chapter 1: Relations and Functions
Q: 10 | Page no. 19
Solution 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) = Iy(Y) = Fog2(Y). Use One-one Ness of F). Concept: Composition of Functions and Invertible Function.
S