Question

Find the domain of the function.

$f\left(x\right)=\frac{x}{\sqrt[4]{9-x^2}}$

Short answer

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

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:

$f\left(x\right)=\frac{x}{\sqrt[4]{9-x^2}}$

Step 3. Domain of the given function.

The given function is a rational function. The numerator is $x$. It is defined for all real values of x.

Denominator is defined if the radicand is non negative, i.e.,$9-x^2\ge 0$ but the denominator cannot be equal to $0$, therefore the function is defined if:

role="math" localid="1644605417329" $$\begin{gathered} 9-x^2>0 \\ 9>x^2 \\ {3}^2>x^2 \\ -3<x<3\left[\because a^2>x^2\Rightarrow -a<x<a\right] \end{gathered}$$

The given function is defined for all real values of$x$ that are greater than$-3$ and less than $3$.

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