Question

The domain of the function $f\left(x\right)=\frac{1+2{\left(x+4\right)}^{-0.5}}{2-{\left(x+4\right)}^{0.5}}+5{\left(x+4\right)}^{0.5}$ is

• A. $R$
• B. $(-4,4)$
• C. $\left(0,\infty \right)$
• D. $\left(-4,0\right)\cup \left(0,\infty \right)$

Answer 1

Answer: (D)

$f\left(x\right)=\frac{1+2{\left(x+4\right)}^{-0.5}}{2-{\left(x+4\right)}^{0.5}}+5{\left(x+4\right)}^{0.5}$
$=\frac{1+2\cdot \frac{1}{\sqrt{x+4}}}{2-\sqrt{x+4}}+5\sqrt{x+4}=\frac{\sqrt{x+4}+2}{\sqrt{x+4}\left(2-\sqrt{x+4}\right)}+5\sqrt{x+4}$
$f\left(x\right)$ is defined if $x+4>0$ and $2-\sqrt{x+4}\ne 0$.
$\therefore f\left(x\right)$ is defined if $x>-4$ and $x\ne 0$
$\therefore D\left(f\right)=\left(-4,0\right)\cup \left(0,\infty \right)$.