Question:
$f: R \rightarrow R$ is defined by $f(x)=\frac{e^{x^{2}}-e^{-x^{2}}}{e^{x^{2}+e^{-x^{2}}}}$ is
(a) one-one but not onto
(b) many-one but onto
(c) one-one and onto
(d) neither one-one nor onto
Solution:
(d) neither one-one nor onto
We have,
$f(x)=\frac{e^{x^{2}}-e^{-x^{2}}}{e^{x^{2}+e^{-x^{2}}}}$
Here, $-2,2 \in R$
Now, $2 \neq-2$
But, $f(2)=f(-2)$
But, $f(2)=f(-2)$
Therefore, function is not one - one.
And,
The minimum value of the function is 0 and maximum value is 1
That is range of the function is $[0,1]$ but the co-domain of the function is given R.
Therefore, function is not onto.
$\therefore$ function is neither one $-$ one nor onto.