Question

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

Answer 1

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

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

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

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

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

Now x will be real if:

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

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

$\Rightarrow 0\le y<1$

$\Rightarrow yϵ[0,1)$

$\therefore$ Range of f(x) = [0, 1)