# In Each of the Following Cases, State Whether the Function is One-one, onto Or Bijective. Justify Your Answer. F: R → R Defined By F(X) = 1 + X2 - Mathematics

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

f→ R defined by f(x) = 1 + x2

#### Solution

fR → R is defined as

f(x) = 1 + x^2

Let x_1, x_2  in "R such that " f(x_1) = f(x_2)

=> 1 + x_1^2 = 1 + x_2^2

=> x_1^2 = x_2^2

=> x_1 = +-x_2

∴ f(x_1) = f(x_2) does not imply that x_1 = x_2

For instance,

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.

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.

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#### APPEARS IN

NCERT Class 12 Maths
Chapter 1 Relations and Functions
Q 7.2 | Page 11