Functions
Surjective Functions
JEE Main 2019
Grade 12
Question:
$f : \mathbb{R} - \{1, -1\} \to A$ defined by $f(x) = \frac{x^2}{1 - x^2}$ is surjective. Then $A$ equals:
(1) $\mathbb{R} - \{-1\}$
(2) $\mathbb{R} - [-1, 0)$
(3) $\mathbb{R} - (-1, 0)$
(4) $[0, \infty)$
Step-by-Step Solution
Key Concept: Set $y = x^2/(1 - x^2)$ and solve for $x^2$. For $x^2 \geq 0$, determine which values of $y$ are achievable. The values $y \in [-1, 0)$ cause problems — identify why.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)