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Give Examples of Two Functions F: N → Z And G: Z → Z Such That G O F Is Injective But Gis Not Injective. (Hint: Consider F(X) = X And G(X) =|X|) - CBSE (Commerce) Class 12 - Mathematics

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Question

Give examples of two functions fN → Z and gZ → Z such that g o f is injective but gis not injective.

(Hint: Consider f(x) = x and g(x) =|x|)

Solution

Define fN → Z as f(x) = x and gZ → Z as g(x) =|x|.

We first show that g is not injective.

It can be observed that:

g(−1) = `|-1| = 1`

g(1) =  `|1| = 1`

∴ g(−1) = g(1), but −1 ≠ 1.

∴ g is not injective.

Now, gofN → Z is defined as

`gof(x) = g(f(x)) = g(x) = |x|`

Let xy ∈ N such that gof(x) = gof(y).

⇒ |x| = |y|

Since x and y ∈ N, both are positive.

`:. |x| = |y| => x = y`

Hence, gof is injective

  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: 6 | Page no. 29
Solution Give Examples of Two Functions F: N → Z And G: Z → Z Such That G O F Is Injective But Gis Not Injective. (Hint: Consider F(X) = X And G(X) =|X|) Concept: Types of Functions.
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