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).