A function $f: R→ R$ defined by $f(x) =\frac{x}{x^2+1}$, is (where R is a set of real number)

neither one-one nor onto

The correct answer is Option (3) → neither one-one nor onto

Given function: $f(x)=\frac{x}{x^2+1}$

Check if the function is one–one:

To be one–one, different inputs must give different outputs. But here:

$f(2)=\frac{2}{5}$

$f\left(\frac{1}{2}\right)=\frac{\frac{1}{2}}{\frac{1}{4}+1} =\frac{\frac{1}{2}}{\frac{5}{4}} =\frac{2}{5}$

Since $2 \neq \frac{1}{2}$ but $f(2)=f\left(\frac{1}{2}\right)$, the function is not one–one.

Check if the function is onto ($R\to R$):

Range calculation gives:

$|y| \le \frac{1}{2}$

So the range is $\left[-\frac{1}{2},\frac{1}{2}\right]$, which is not all real numbers.

Therefore, the function is not onto.

Hence, the function is neither one–one nor onto.