Suppose that the functions \( f \) and \( g \) are defined as follows. \[ f(x)=\frac{x-9}{x+5} \quad g(x)=\frac{x}{x+5} \] Find \( \frac{f}{g} \), Then, give its domain using an interval or union of interval Simplify your answers. \( \left(\frac{f}{g}\right)(x)= \) II Domain of \( \frac{f}{g} \) :

Given the functions \( f(x) = \frac{x-9}{x+5} \) and \( g(x) = \frac{x}{x+5} \), we need to find the function \( \frac{f}{g} \) and then determine its domain. To find \( \frac{f}{g} \), we divide the function \( f(x) \) by the function \( g(x) \): \[ \frac{f}{g} = \frac{\frac{x-9}{x+5}}{\frac{x}{x+5}} \] To simplify this expression, we can cancel out the common factor \( (x+5) \) in the numerator and denominator: \[ \frac{f}{g} = \frac{x-9}{x} \] Therefore, the function \( \frac{f}{g} \) is \( \frac{x-9}{x} \). To determine the domain of \( \frac{f}{g} \), we need to consider the values of \( x \) for which the function is defined. Since the function is a quotient of two functions, it is defined for all values of \( x \) except where the denominator is zero. The denominator of \( \frac{f}{g} \) is \( x \), so the function is undefined when \( x = 0 \). Therefore, the domain of \( \frac{f}{g} \) is all real numbers except \( x = 0 \), which can be represented as \( (-\infty, 0) \cup (0, \infty) \).