Question

The domain of $f\left(x\right)={\log }_5(x^2+4x+1)$ is

• A. $(-\infty ,-2-\sqrt{3})$
• B. $(-2+\sqrt{3},\infty )$
• C. $(-2-\sqrt{3},-2+\sqrt{3})$
• D. $(-\infty ,-2-\sqrt{3})\cup (-2+\sqrt{3},\infty )$

Answer 1

The correct option is D $(-\infty ,-2-\sqrt{3})\cup (-2+\sqrt{3},\infty )$

For $f\left(x\right)={\log }_5(x^2+4x+1)$ to be defined, we get
$x^2+4x+1>0\Rightarrow x^2+4x+4-3>0\Rightarrow (x+2{)}^2>3\Rightarrow x+2>\sqrt{3},x+2<-\sqrt{3}\Rightarrow x\in (-\infty ,-2-\sqrt{3})\cup (-2+\sqrt{3},\infty )$