In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i). f : R → R defined by f (x) = 3 - 4x (ii). f : R → R defined by f (x) = 1 + x²

Solution:

i.

f : R → R defined by f (x) = 3 - 4x

x₁, x₂ ∈ R such that f (x₁) = f (x₂)

⇒ 3 - 4x₁ = 3 - 4x₂

⇒ - 4x = - 4x₂

⇒ x₁ = x₂

⇒ f is one-one.

For any real number (y) in R,

there exists (3 - y)/4 in R such that

f (3 - y)/4 = 3 - 4 (3 - y)/4

= y

⇒ f is onto.

Hence, f is bijective.

ii.

f : R → R defined by f (x) = 1 + x²

x₁, x₂ ∈ R such that f (x₁) = f (x₂)

⇒ 1 + x ² = 1 + x ²

⇒ x² = x²

⇒ x₁ = ±  x₂

⇒ f ( x₁) = f (x₂)

does not imply that x₁ = x₂

Consider f (1) = f (- 1) = 2

⇒ f is not one-one.

Consider an element -2 in co-domain R.

It is seen that f (x) = 1 + x² is positive for all x ∈ R.

⇒ f is not onto.

Hence, f is neither one-one nor onto

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.2 Question 7

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i). f : R → R defined by f (x) = 3 - 4x (ii). f : R → R defined by f (x) = 1 + x²

Summary:

Given that (i) f : R → R defined by f (x) = 3 - 4x. F is one - one as well as onto hence it is bijective. (ii) f is neither one-one nor onto