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.
✓
Answer
Here $f ( x ) = \frac { x ^ { 2 } } { 1 + x ^ { 2 } }$
Put $y = \frac { x ^ { 2 } } { 1 + x ^ { 2 } } \Rightarrow y + yx^2 = x^2 \Rightarrow x^2(1 - y) = y$
$\Rightarrow x ^ { 2 } = \frac { y } { 1 - y } \Rightarrow x = \pm \sqrt { \frac { y } { 1 - y } }$
$\frac { y } { 1 - y } \geq 0$
$\Rightarrow \frac { y } { y - 1 } \leq 0$
$\Rightarrow 0 \leq \mathrm { y } < 1$
$\Rightarrow \mathrm { y } \in [ 0,1 )$
$\therefore$ Range of $f(x) = [0, 1)$
Put $y = \frac { x ^ { 2 } } { 1 + x ^ { 2 } } \Rightarrow y + yx^2 = x^2 \Rightarrow x^2(1 - y) = y$
$\Rightarrow x ^ { 2 } = \frac { y } { 1 - y } \Rightarrow x = \pm \sqrt { \frac { y } { 1 - y } }$
$\frac { y } { 1 - y } \geq 0$
$\Rightarrow \frac { y } { y - 1 } \leq 0$
$\Rightarrow 0 \leq \mathrm { y } < 1$
$\Rightarrow \mathrm { y } \in [ 0,1 )$
$\therefore$ Range of $f(x) = [0, 1)$
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