Question

The domain of the function $f\left(x\right)={\log }_{3+x}\left(x^2-1\right)$ is

• A. $\left(-3,-1\right)\cup \left(1,\infty \right)$
• B. $[-3,-1)\cup [1,\infty )$
• C. $\left(-3,-2\right)\cup \left(-2,-1\right)\cup \left(1,\infty \right)$
• D. $[-3,-1)\cup \left(-2,-1\right)\cup [1,\infty )$

Answer 1

The correct option is C

$\left(-3,-2\right)\cup \left(-2,-1\right)\cup \left(1,\infty \right)$

Explanation for correct answer Option(s)

Find the domain of the function $f\left(x\right)$

Given function, $f\left(x\right)={\log }_{3+x}\left(x^2-1\right)$

We know that ${\log }_{a}x$ is defined as $a\ne 1$, $a>0$ and $x>0$.

Applying $a\ne 1$on the given function,

$$\begin{gathered} \Rightarrow 3+x\ne 1 \\ \Rightarrow x\ne -2 \\ \therefore x\ne -2 \end{gathered}$$

Applying $a>0$ on the given function,

$$\begin{gathered} \Rightarrow 3+x>0 \\ \Rightarrow x>-3 \\ \therefore x>-3 \end{gathered}$$

Applying $x>0$ on the given function,

$$\begin{gathered} \Rightarrow x^2-1>0 \\ \Rightarrow \left(x+1\right)\left(x-1\right)>0 \\ \therefore x\in \left(-\infty ,-1\right)\cup \left(1,\infty \right) \end{gathered}$$

By comparing the above three, the domain of the given function would be $\left(-3,-2\right)\cup \left(-2,-1\right)\cup \left(1,\infty \right)$.

Therefore, the correct Option is (C).