Question

Domain Find the domain of the function.

$g\left(x\right)=\sqrt{x^2-2x-8}$

Short answer

The domain of the given function is $\left(-\infty ,-2\right]\cup \left[4,\infty \right)$.

Step-by-step solution

Step 1. Definition of domain.

The set of all input values of a function is known as the domain of the function.

Step 2. Given information.

The given function is:

$g\left(x\right)=\sqrt{x^2-2x-8}$

This function can be rewritten as:

$$\begin{gathered} g\left(x\right)=\sqrt{x^2-(4-2)x-8} \\ =\sqrt{x^2-4x+2x-8} \\ =\sqrt{x(x-4)+2(x-4)} \\ =\sqrt{(x+2)(x-4)} \end{gathered}$$

Step 3. Domain of the given function.

The given function is defined if the radicand is greater than or equal to $0$.

$\left(x+2\right)\left(x-4\right)\ge 0$

For $\left(x+2\right)\left(x-4\right)=0$, $x+2=0$and $x-4=0$. So, the values of $x$are $-2$and $4$respectively.

Two number $-2$and $4$divide the number line in three intervals $\left(-\infty ,-2\right),\left(-2,4\right)$and $\left(4,\infty \right)$.

The table of signs is:

Interval	localid="1644602093720" $\left(-\infty ,-2\right)$	localid="1644602100440" $\left(-2,4\right)$	$\left(4,\infty \right)$
Sign of localid="1644602131288" $(x+2)$	-	+	+
Sign of localid="1644602124523" $(x-4)$	-	-	+
Sign of localid="1644602116457" $(x+2)(x-4)$	+	-	+

The value of$\left(x+2\right)\left(x-4\right)$ is positive for the intervals$\left(-\infty ,-2\right)$ and $\left(4,\infty \right)$. The values$-2$ and$4$ are included in the domain.

Thus, the domain of the given function is $\left(-\infty ,-2\right]\cup \left[4,\infty \right)$.