The function f : R→R,
Question:

The function $f: R \rightarrow R, f(x)=x^{2}$ is

(a) injective but not surjective
(b) surjective but not injective
(c) injective as well as surjective
(d) neither injective nor surjective

 

Solution:

Injectivity:
Let x and y be any two elements in the domain (R), such that f(x) = f(y). Then,

$x^{2}=y^{2}$

$\Rightarrow x=\pm y$

So, f is not one-one.

Surjectivity:

As $f(-1)=(-1)^{2}=1$

and $f(1)=1^{2}=1$,

$f(-1)=f(1)$

So, both $-1$ and 1 have the same images.

$\Rightarrow f$ is not onto.

So, the answer is (d).

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