Let f: R → R be defined by f(x) = x/(1+x^2), x ∈ R. Then, the range of f is

Let f: R → R be defined by f(x) = x/(1+x^2), x ∈ R. Then,… — JEE Main 2019 Mathematics PYQ

JEE Main 2019 · 11 Jan · Shift I

Let

f: R \to R

be defined by

f(x) = \dfrac{x}{1+x^2}

x \in R

. Then, the range of

f

A.
$\left[-\dfrac{1}{2}, \dfrac{1}{2}\right]$
B.
$(-1, 1) - \{0\}$
C.
$R - \left[-\dfrac{1}{2}, \dfrac{1}{2}\right]$
D.
$R - [-1, 1]$

Show answer and solution

Correct option: A

Solution

Set y = x/(1+x²), then yx² - x + y = 0. For real x, the discriminant must be ≥ 0: 1 - 4y² ≥ 0, giving y² ≤ 1/4, so -1/2 ≤ y ≤ 1/2. Both endpoints are achieved (y = 1/2 at x = 1, y = -1/2 at x = -1), so the range is exactly [-1/2, 1/2].

[JEE Main 2019 · 11 Jan · Shift I]

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