Question

The domain of definition of $f\left(x\right)=\sqrt{\frac{x+3}{(2-x)(x-5)}}$ is

• A. $(\infty ,-3)\cup (2,5)$
• B. $(-\infty ,3)\cup (2,5)$
• C. $(-\infty ,-3)\cup (2,5)$
• D. None of these

Answer 1

The correct option is C

$(-\infty ,-3)\cup (2,5)$

$f\left(x\right)=\sqrt{\frac{x+3}{(2-x)(x-5)}}$
For f(x) to be defined,
$(2-x)(x-5)\ne 0\Rightarrow x\ne 2,5\dots \dots \left(1\right)$
Also, $\frac{(x+3)}{(2-x)(x-5)}\ge 0$
$\Rightarrow \frac{(x+3)(2-x)(x-5)}{(2-x{)}^2(x-5{)}^2}\ge 0\Rightarrow (x+3)(x-2)(x-5)\le 0\Rightarrow xϵ(-\infty ,-3]\cup (2,5)\dots \dots \left(2\right)$
From (1) and (2),
$xϵ(-\infty ,-3]\cup (2,5)$