Suppose that the functions \( f \) and \( g \) are defined as follows. \[ f(x)=\frac{6}{x+2} \quad g(x)=\frac{x}{x-3} \] Find \( \frac{g}{f} \). Then, give its domain using an interval or union of intervals. Simplify your answers. \( \left(\frac{g}{f}\right)(x)=\square \) Domain of \( \frac{g}{f}: \square \)

To find \( \frac{g}{f} \), we will divide the function \( g(x) \) by \( f(x) \). 1. **Write down the functions:** \[ f(x) = \frac{6}{x+2}, \quad g(x) = \frac{x}{x-3} \] 2. **Calculate \( \frac{g}{f} \):** \[ \frac{g}{f}(x) = \frac{g(x)}{f(x)} = \frac{\frac{x}{x-3}}{\frac{6}{x+2}} \] 3. **Simplify the expression:** To divide by a fraction, we multiply by its reciprocal: \[ \frac{g}{f}(x) = \frac{x}{x-3} \cdot \frac{x+2}{6} = \frac{x(x+2)}{6(x-3)} \] Thus, we have: \[ \left(\frac{g}{f}\right)(x) = \frac{x(x+2)}{6(x-3)} \] 4. **Determine the domain of \( \frac{g}{f} \):** The domain is restricted by the points where the denominator is zero. We need to find when \( f(x) \) and \( g(x) \) are undefined. - \( f(x) \) is undefined when \( x + 2 = 0 \) or \( x = -2 \). - \( g(x) \) is undefined when \( x - 3 = 0 \) or \( x = 3 \). Therefore, the function \( \frac{g}{f}(x) \) is undefined at \( x = -2 \) and \( x = 3 \). 5. **Express the domain:** The domain of \( \frac{g}{f} \) in interval notation is: \[ (-\infty, -2) \cup (-2, 3) \cup (3, \infty) \] In conclusion, we have: \[ \left(\frac{g}{f}\right)(x) = \frac{x(x+2)}{6(x-3)} \] Domain of \( \frac{g}{f}: (-\infty, -2) \cup (-2, 3) \cup (3, \infty) \)