Question

The domain of the function
$f\left(x\right)=x\cdot \frac{1+2(x+4{)}^{-0.5}}{2-(x+6{)}^{0.5}}+(x+5{)}^{0.5}+4(x+10{)}^{-0.5}$ is

• A. $(-4,\infty )-\left\{2\right\}$
• B. $(-4,\infty )-\{-2\}$
• C. $R-\left(-4,0\right)$
• D. $(0,\infty )$

Answer 1

The correct option is B $(-4,\infty )-\{-2\}$

$f\left(x\right)=x\cdot \frac{1+2(x+4{)}^{-0.5}}{2-(x+6{)}^{0.5}}+(x+5{)}^{0.5}+4(x+10{)}^{-0.5}\Rightarrow f\left(x\right)=x\cdot \frac{1+\frac{2}{\sqrt{x+4}}}{2-\sqrt{(x+6)}}+\sqrt{x+5}+\frac{4}{\sqrt{x+10}}$

For the function to be defined
$x+4>0,x+6\ge 0,x+5\ge 0,x+10>0,2-\sqrt{x+6}\ne 0$
From the above conditions, we get
$\Rightarrow x>-4,\sqrt{x+6}\ne 2\Rightarrow x>-4,x\ne -2$
Hence, the required domain is
$(-4,\infty )-\{-2\}$