Question

For real $x$, let $f(x) = x^3 + 5x + 1$. Then :

• A. $f$ is one-one but not onto on $\mathbb{R}$
• B. $f$ is onto on $\mathbb{R}$ but not one-one
• C. $f$ is one-one and onto on $\mathbb{R}$
• D. $f$ is neither one-one nor onto on $\mathbb{R}$

Hint:

To check whether a function is one-one or onto, always check its derivative. If the derivative is always positive or negative, the function is monotonic and thus one-one.

Answer 1

The Correct Option is C

To determine whether the function $f(x) = x^3 + 5x + 1$ is one-one and onto, we first examine its derivative to check for monotonicity: \[ f'(x) = 3x^2 + 5 \] Since $f'(x) = 3x^2 + 5>0$ for all $x$, $f(x)$ is strictly increasing and hence one-one. Additionally, since the function is strictly increasing, it is also onto $\mathbb{R}$ as it can take any real value. Hence, the function is both one-one and onto on $\mathbb{R}$.