For which real numbers $a$ is the function $f (x) = x^{3} + a x$ one‑to‑one on $R$ ? Justify your answer.

hard 3 marks Class 12 · Mathematics · Relations and Functions

Answer

a \geq 0

The correct answer is $a \geq 0$ — If $a < 0$ , then $f^{'} (0) = a < 0$ and $f^{'}$ changes sign, making $f$ non‑monotonic; thus $f$ is not one‑to‑one.

Step-by-step solution

Compute derivative: $f^{'} (x) = 3 x^{2} + a$ .
The minimum of $f^{'}$ occurs at $x = 0$ and equals $a$ .
If $a \geq 0$ , then $f^{'} (x) = 3 x^{2} + a \geq a \geq 0$ for all $x$ , so $f$ is strictly increasing and hence one‑to‑one.
If $a < 0$ , then $f^{'} (0) = a < 0$ and $f^{'}$ changes sign, making $f$ non‑monotonic; thus $f$ is not one‑to‑one.

Common mistake: Assuming all odd functions are injective regardless of linear term

Concept tested: Conditions for injectivity of cubic functions

Published 10 October 2026 · How this solution was prepared and checked

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For which real numbers a is the function f(x)=x3+ax one‑to‑one on R? Justify your answer.

Step-by-step solution

For which real numbers $a$ is the function $f (x) = x^{3} + a x$ one‑to‑one on $R$ ? Justify your answer.