Question

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

Answer 1

Let $y=\frac{x^2}{1+x^2}$

$\Rightarrow y+x^{2}y=x^2$

$\Rightarrow y=x^2(1-y)$

$\Rightarrow x^2=\frac{y}{1-y}$

$\Rightarrow x=\sqrt{\frac{y}{1-y}}$

Since, $x$ is real

$\Rightarrow \frac{y}{1-y}\ge 0$

$\Rightarrow \frac{y(1-y)}{(1-y{)}^2}\ge 0$

$\Rightarrow y(1-y)\ge 0$ and $(1-y{)}^2>0$

$\Rightarrow 0\le y\le 1$ and $-y>-1$

$\Rightarrow 0\le y\le 1$ and $y<1$

Hence, $0\le y<1$

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