Question

The set of all $x$ for which the functions are not defined
$f\left(x\right)={\log }_{\left[\right(x-2\left)/\right(x+3\left)\right]}2$ and $g\left(x\right)=\frac{1}{\sqrt{x^2-9}}$, is

• A. $[-3,2]$
• B. $[-3,2)$
• C. $(-3,2]$
• D. $(-3,-2)$

Answer 1

Answer: (A)

$[-3,2]$

$g\left(x\right)=\frac{1}{\sqrt{x^2-9}}$

$x^2-9>0$

$x>3,x<-3$

but we require

$-3\le x\le 3$

$\Rightarrow x\in [-3,3]$ as $g\left(x\right)$ will not be defined in this interval

$f\left(x\right)=\log \left(\frac{x-2}{x+3}\right)$

$\frac{x-2}{x+3}>0,1$

$x>2$

but we require

$x\le 2$

$\Rightarrow x\in [-\infty ,2]$ as $f\left(x\right)$ will not be defined in this interval

Taking common of both interval we get

$-3\le x\le 2$

$\therefore x\in [-3,2]$

From given options option A is appropriate