Answer
The domain of this function is $(-3,3)$
Work Step by Step
$f(x)=\dfrac{x}{\sqrt[4]{9-x^{2}}}$
This function is undefined for negative values of the expression inside the square root and also for the values of $x$ that make the denominator equal to $0$. Its domain can then be found by solving the following inequality:
$9-x^{2}\gt0$
Find the intervals. Factor the left side of the inequality:
$(3-x)(3+x)\gt0$
The factors are $3-x$ and $3+x$. Set them equal to $0$ and solve for $x$:
$3-x=0$
$x=3$
$3+x=0$
$x=-3$
The factors are $0$ when $x=3,-3$. These numbers divide the real into the following intervals:
$(-\infty,-3)$ $,$ $(-3,3)$ $,$ $(3,\infty)$
Elaborate a diagram, using test points to determine the sign of each factor in each interval: (refer to the attached image below)
It can be seen from the diagram that only the interval $(-3,3)$ satisfies the inequality. Also, since it involves the sign $\gt$, the endpoints of this interval do not satisfy the inequality.
The domain of this function is $(-3,3)$
