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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

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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.

  Is there an error in this question or solution?
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APPEARS IN

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