Suppose that the functions \( f \) and \( g \) are defined as follows. \[ f(x)=\frac{2}{x} \quad g(x)=x-9 \] Find \( \frac{f}{g} \). Then, give its domain using an interval or union of intervals. Simplify your answers. \( \left(\frac{f}{g}\right)(x)=\square \) Domain of \( \frac{f}{g}: \square \)

To find \( \frac{f}{g} \), we calculate: \[ \frac{f}{g}(x) = \frac{f(x)}{g(x)} = \frac{\frac{2}{x}}{x - 9} = \frac{2}{x(x - 9)} \] Next, we need to determine the domain of \( \frac{f}{g} \). The function is undefined whenever the denominator is zero. Thus, we set the denominator to zero: \[ x(x - 9) = 0 \] This gives us the solutions \( x = 0 \) and \( x = 9 \). Therefore, we exclude these values from the domain of the function. The domain is all real numbers except \( 0 \) and \( 9 \). Thus, we can represent the domain in interval notation: Domain of \( \frac{f}{g}: (-\infty, 0) \cup (0, 9) \cup (9, \infty) \). So, the final answers are: \[ \left(\frac{f}{g}\right)(x)=\frac{2}{x(x - 9)} \] Domain of \( \frac{f}{g}: (-\infty, 0) \cup (0, 9) \cup (9, \infty) \).