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In the following case, state whether the function is one-one, onto or bijective. Justify your answer. f : R → R defined by f(x) = 1 + x^2

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Question

In the following case, state whether the function is one-one, onto or bijective. Justify your answer.

f : R → R defined by f(x) = 1 + x2

Justify
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Solution

f : R → R is defined by f(x) = 1 + x2

Let x1, x2 ∈ R such that f(x1) = f(x2)

⇒ `1 + x_1^2 = 1 + x_2^2`

⇒ `x_1^2 = x_2^2`

⇒ x1 = ±x2

∴ f(x1) = f(x2) does not imply that x1 = x2.

For instance, f(1) = f(–1) = 2

 ∴ f is not one-one.

Consider the element –2 in co-domain R.

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

Thus, there does not exist any x in domain R such that f(x) = –2.

∴ f is not onto.

Hence, f is neither one-one nor onto and hence not bijective.

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Chapter 1: Relations and Functions - EXERCISE 1.2 [Page 11]

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NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 1 Relations and Functions
EXERCISE 1.2 | Q 7. (ii) | Page 11

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