Functions
Range of a Function
JEE Main 2019
Grade 12
Question:
The range of the function $f(x) = \frac{x}{1 + x^2}$, $x \in \mathbb{R}$, is:
(1) $(-1, 1)$
(2) $[-1, 1]$
(3) $[-\frac{1}{2}, \frac{1}{2}]$
(4) $(-\frac{1}{2}, \frac{1}{2})$
Step-by-Step Solution
Key Concept: Set $y = x/(1 + x^2)$ and rearrange to $yx^2 - x + y = 0$. For real $x$, the discriminant must be $\geq 0$. Find the constraint on $y$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)