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 verify if the function $f(x) = x^3 + 5x + 1$ is one-one and onto, we analyze its derivative for monotonicity: \[f'(x) = 3x^2 + 5\] Because $f'(x) = 3x^2 + 5>0$ for all real numbers $x$, the function $f(x)$ is strictly increasing, which implies it is one-one. Furthermore, due to its strictly increasing nature, the function is also onto $\mathbb{R}$, as it spans all real values. Consequently, the function is both one-one and onto on $\mathbb{R}$.