f : R→R is defined by f(x)

(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.