Question

Let $f=\left\{\left(x,\frac{x^2}{1+x^2}\right):x\in R\right\}$ be a function from $R$ into $R$. The range of $f$ is

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

Answer 1

Answer: (B)

We have, $f=\left\{\left(x,\frac{x^2}{1+x^2}\right),x\in R\right\}$.
Clearly, domain of $f$ is $R$.
Let $y=\frac{x^2}{1+x^2}\cdot$ It is clear that $\frac{x^2}{1+x^2}\ge 0$
$(\because x^2\ge 0$ and $1+x^2\ge 0)$
and $x^2<1+x^2$
$\Rightarrow \frac{x^2}{1+x^2}<1$
So this implies $0\le y<1$.
Hence, range of $f$ is $[0,1)$.