# 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 + x^{2}

**Solution:**

i.

f : R → R defined by f (x) = 3 - 4x

x_{1}, x_{2} ∈ R such that f (x_{1}) = f (x_{2})

⇒ 3 - 4x_{1} = 3 - 4x_{2}

⇒ - 4x = - 4x_{2}

⇒ x_{1} = x_{2}

⇒ 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 + x^{2}

x_{1}, x_{2} ∈ R such that f (x_{1}) = f (x_{2})

⇒ 1 + x ^{2} = 1 + x ^{2}

⇒ x^{2} = x^{2}

⇒ x_{1} = ± x_{2}

⇒ f ( x_{1}) = f (x_{2})

does not imply that x_{1} = x_{2}

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 + x^{2} is positive for all x ∈ R.

⇒ f is not onto.

Hence, f is neither one-one nor onto

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 + x^{2}

**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