Question

Find the domain of definition and the range of the following function:
$f\left(x\right)={\log }_{\frac{x-2}{x+3}}2$ and $g\left(x\right)=\sqrt{x^2-9}.$

Answer 1

${\log }_{\frac{x-2}{x+3}}2=\frac{\log 2}{\log \frac{x-2}{x+3}}\Rightarrow \frac{x-2}{x+3}>0(x-2)(x+3)>0x\ne -3x\in (-\infty ,-3)\cup (2,\infty )$

Domain of $x\in (-\infty ,-3)\cup (2,\infty )$

Range of $y\in (0,\infty )$

$g\left(x\right)=\sqrt{x^2-9}{x}^2-9\ge 0x\in (-\infty ,-3]\cup [3,\infty )$

Domain of $x\in (-\infty ,-3]\cup [3,\infty )$

Range of $y\in [0,\infty )$