Question

The largest set of real values of $x$ for which $f\left(x\right)=\sqrt{(x+2)(5-x)}-\frac{1}{\sqrt{x^2-4}}$ is a real function is

• A. $[1,2)\cup (2,5]$
• B. $(2,5]$
• C. $[3,4]$
• D. $noneofthese$

Answer 1

The correct option is B $(2,5]$

$\sqrt{(x+2)(5-x)}$ is defined for

$(x+2)(5-x)\ge 0\Rightarrow (x+2)(x-5)\le 0\Rightarrow x\in [-2,5].........\left(1\right)$
.
Similarly, $\frac{1}{\sqrt{x^2-4}}$ is defined for

$x^2-4>0x\in (-\infty ,-2)\bigcup (2,\infty )......\left(2\right)$
.
From $\left(1\right)$ and $\left(2\right)$, the domain of $f\left(x\right)$ is $x\in (2,5]$