Give examples of two functions f: N → Z and g: Z → Z such that gof is injective but g is not injective

Solution:

In mathematics, an injective function ( or one-to-one function) is a function f such that maps distinct elements to distinct elements;

that is, f(x₁) = f (x₂) 

⇒ x₁ = x₂

Define f : N → Z as f (x) = x and

g : Z → Z as g (x) = |x|

Let us first show that g is not injective.

(- 1) = |- 1| = 1

(1) = |1| = 1

⇒ (- 1) = g (1), but - 1 ≠ 1

⇒ g is not injective.

gof : N → Z is defined as gof (x) = g (f (x)) = g (x) = |x|

x, y ∈ N such that gof (x) = gof (y)

⇒ |x| = |y|

Since x, y ∈ N, both are positive.

⇒ |x| = |y|

⇒ x = y

Therefore,

gof is injective

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise ME Question 6

Give examples of two functions f: N → Z and g: Z → Z such that gof is injective but g is not injective.

Summary:

f: N → Z as f (x) = x and g: Z → Z as g (x) = |x| are the examples of two functions f: N → Z and g: Z → Z such that gof is injective but g is not injective