Find domain of the quotient of two square root functions

Question

Question: Given $$f(x)=\sqrt{36-x^2} \\ g(x)=\sqrt{x+1}$$ then find $$\left(\frac fg\right)(x) = \sqrt{\frac{36-x^2}{x+1}}$$ and find its domain.

My Solution: The domain comes from $$x+1\ne0$$ and (intersection) $$\frac{36-x^2}{x+1}\ge0.$$ The first part implies that $x\ne-1$. But $$\frac{36-x^2}{x+1} \ge 0$$ when $x$ is in $(-\infty,-6] \cup (-1,6].$

So my final answer is $(-\infty,-6] \cup (-1,6]$.

WebAssign's Answer: $(-1,6]$.

Answer 1

Because the new function is a quotient of two functions, the domains of the original functions need to be taken into account too. In this case $f(x)$ has domain $[-6,6]$ and $g(x)$ has domain $[-1,\infty)$, so these needs to be intersected with the calculated domain $(-\infty,-6]\cup(-1,6]$ to get the correct answer of $(-1,6]$.

Answer 2

The domain of $\frac{f}{g}$ is $x+1>0$ and $36-x^2\geq0,$ which is

$x>-1$ and $-6\leq x\leq6$ or

$-1<x\leq6,$ which gives the answer: $(-1,6]$.