Find the range of the following function given by: f(x)=3 2-x^2 - Vidyadip

Question

Find the range of the following function given by:
$\text{f(x)}=\frac{3}{2-\text{x}^2}$

✓

Answer

We have, $\text{f(x)}=\frac{3}{2-\text{x}^2}=\text{y}$ (let)
$\Rightarrow2-\text{x}^2=\frac{3}{\text{y}}$
$\Rightarrow\text{x}^2=2-\frac{3}{\text{y}}$
Since $\text{x}^2\geq0,2-\frac{3}{\text{y}}\geq0$
$\Rightarrow\frac{2\text{y}-3}{\text{y}}\geq0$
$\Rightarrow2\text{y}-3\geq0\text{y}>0\Rightarrow2\text{y}-3\leq0\text{y}<0$
$\Rightarrow\text{y}\geq\frac{3}{2}\text{y}<0$
$\Rightarrow\text{y}\in(-\infty, 0)\cup[\frac{3}{2},\infty)$
$\therefore$ Range of f is $(-\infty,0)\cup[\frac{3}{2}, \infty)$

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