# find the domain and range of 3/2-x^2

the given function is : $y=f\left(x\right)=\frac{3}{2-{x}^{2}}$
since denominator can not be zero:

therefore domain is $R-\left\{-\sqrt{2},\sqrt{2}\right\}$
for range:
$y=\frac{3}{2-{x}^{2}}\phantom{\rule{0ex}{0ex}}2y-{x}^{2}y=3\phantom{\rule{0ex}{0ex}}{x}^{2}y=2y-3\phantom{\rule{0ex}{0ex}}{x}^{2}=\frac{2y-3}{y}$
since square of any number is positive:
$\frac{2y-3}{y}\ge 0\phantom{\rule{0ex}{0ex}}⇒y\in \left(-\infty ,0\right)\cup \left[\frac{3}{2},\infty \right)$
hope this helps you.

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