The function $f: \mathbf{R} \rightarrow\left[-\frac{1}{2}, \frac{1}{2}\right]$ defined as $f(x)=\frac{x}{1+x^{2}}$, is :

(D) surjective but not injective.

(A) neither injective nor surjective.

(C) injective but not surjective.

JEE Main · 2017 · 02 Apr 2017 · Offline · Sets Relations and Functions

The function

f : R \to [- \frac{1}{2}, \frac{1}{2}]

defined as

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

, is :

Answer: (D)

Differentiating,

f^{'} (x) = \frac{1 - x ^{2}}{( 1 + x ^{2} ) ^{2}}

, positive for

∣ x ∣ < 1

and negative for

∣ x ∣ > 1

, so the function rises then falls and cannot be one-to-one. But its values sweep continuously from the minimum

- \frac{1}{2}

x = - 1

up to the maximum

\frac{1}{2}

x = 1

, covering the whole target interval

[- \frac{1}{2}, \frac{1}{2}]

. So

f

is surjective but not injective.

JEE Main · 2017 · 02 Apr 2017 · Offline · Single correct option