Chapter 8.4, Problem 5QR

To determine

The domain of the function $g\left(x\right)=\frac{1}{\sqrt{9-x^2}}$ .

Answer to Problem 5QR

The domain of the function is $\left(-3,3\right)$ .

Explanation of Solution

Given information: The given function is $g\left(x\right)=\frac{1}{\sqrt{9-x^2}}$ .

Concept used: The domain of the radical function $\sqrt{a}$ is $a\ge 0$ .

Calculation:

Find the domain of the radical $\sqrt{9-x^2}$ .

  $\begin{array}{l}9-x^2\ge 0\\ \Rightarrow 9\ge x^2\\ \Rightarrow -3\ge x\ge 3\end{array}$

At the end points, that is at $x=-3,3$ the value of the radical will be 0.

Since the denominator of the radical function cannot be 0, the end points will not be included in the interval of domain.

So, the domain of the given function is $-3>x>3$ .

Represent the obtained inequality in interval notation.

  $x\in \left(-3,3\right)$

Thus, the domain of the given function is $x\in \left(-3,3\right)$ .