Let f= (x, x^2 1+x^2 ): x R be a function from R into R . Determi… - Vidyadip

Question

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

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Answer

Here $f(x)=\frac{x^2}{1+x^2}$
Put $y=\frac{x^2}{1+x^2} \Rightarrow y+y x^2=x^2 \Rightarrow x^2(1-y)=y$
$\Rightarrow x^2=\frac{y}{1-y} \Rightarrow x= \pm \sqrt{\frac{y}{1-y}}$
$\frac{y}{1-y} \geq 0$
$\begin{array}{l}\Rightarrow \frac{y}{y-1} \leq 0 \\ \Rightarrow 0 \leq y<1 \\ \Rightarrow y \in[0,1)\end{array}$
$\therefore$ Range of $f(x)=[0,1)$

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