Question

If the function $f:R\to A$ given by $f\left(x\right)=\frac{x^2}{x^2+1}$ is a surjection, then A is equal to

• A. R
• B. $[0,1]$
• C. $(0,1]$
• D. $[0,1)$

Answer 1

The correct option is D $[0,1)$

Given that

$f:R\to A$ surjective

$\Rightarrow$ Codomain = range

$\Rightarrow$ range $=A$

$f\left(x\right)=\frac{x^2}{x^2+1},xϵR$

$x^2<x^2+1$

$\Rightarrow \frac{x^2}{x^2+1}<1$ [$\because x^2+1$ is positive]

Also $x^2\ge 0$

$x^2+1>0$

$\Rightarrow \frac{x^2}{x^2+1}\ge 0$

$\therefore$ Range of $f$ is $[0,1)$

$\therefore A$ is $[0,1)$