A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = 2 + x^2$ is:

neither one-to-one nor onto

The correct answer is Option (4) → neither one-to-one nor onto ##

$f(x) = 2 + x^2$

For one-to-one, $f(x_1) = f(x_2)$

$⇒2 + x_1^2 = 2 + x_2^2$

$⇒x_1^2 = x_2^2$

$⇒x_1 = \pm x_2$

Thus, $f(x)$ is not one-to-one.

For onto, Let $f(x) = y$ such that $y \in \mathbb{R}$

$∴x^2 = y - 2$

$⇒x = \pm \sqrt{y - 2}$

Put $y=-3$, we get

$⇒x=\pm\sqrt{-3-2}=\pm\sqrt{-5}$

it is a complex number