Suppose that the functions $f$ and $g$ are defined as follows. \[ f(x)=\frac{1}{x+5} \quad g(x)=\frac{8}{x} \] Find $\frac{f}{g}$. Then, give its domain using an interval or union of intervals. Simplify your answers. \[ \left(\frac{f}{g}\right)(x)= \] Domain of $\frac{f}{g}$ :

Step 1 ：Given the functions \(f(x) = \frac{1}{x+5}\) and \(g(x) = \frac{8}{x}\), we are asked to find \(\frac{f}{g}\) and its domain.

Step 2 ：To find \(\frac{f}{g}\), we divide the function \(f(x)\) by \(g(x)\) to get a new function.

Step 3 ：\(\frac{f}{g}(x) = \frac{\frac{1}{x+5}}{\frac{8}{x}} = \frac{x}{8*(x + 5)}\)

Step 4 ：The simplified form of \(\frac{f}{g}\) is \(\frac{x}{8*(x + 5)}\).

Step 5 ：To find the domain of this function, we need to find all real numbers except for the values that make the denominator equal to zero, because division by zero is undefined.

Step 6 ：The denominator of the function is \(8*x + 40\). The values that make the denominator equal to zero are \(x = -5\) and \(x = 0\).

Step 7 ：Therefore, the domain of the function \(\frac{f}{g}\) is all real numbers except for \(x = -5\) and \(x = 0\).

Step 8 ：\(\boxed{\text{The function } \frac{f}{g} \text{ is } \frac{x}{8*(x + 5)} \text{ and its domain is } (-\infty, -5) \cup (-5, 0) \cup (0, \infty)}\)