For the two functions \( f(x)=9+\frac{7}{x} \) and \( g(x)=\frac{7}{x} \), find each of the needed functions and determine their domains. (For the domains, give your answers as intervals in the form (*,*). Use the symbol \( \infty 0 \) for infinity, \( U \) for combining intervals, and an appropriate type of parenthesis " \( \left.\left("^{\prime \prime}\right)^{\prime}\right)^{\prime}, "\left["^{n},\right]^{\prime} \) " depending on whether the interval is open or closed. Enter \( \varnothing \) if the interval is empty. Express numbers in exact form. Use symbolic notation and fractions where needed.) \[ (f+g)(x)= \] \( \square \) the domain of \( (f+g)(x) \) : \( \square \) \[ (f-g)(x)= \] \( \square \) the domain of \( (f-g)(x) \) : \( \square \) \[ (f \cdot g)(x)= \] \( \square \) the domain of \( (f, g)(x) \) : \( \square \)

Let's analyze the functions \( f(x) = 9 + \frac{7}{x} \) and \( g(x) = \frac{7}{x} \) to find the required operations and their domains. ### 1. Sum of the Functions: \( (f + g)(x) \) \[ (f + g)(x) = \left(9 + \frac{7}{x}\right) + \frac{7}{x} = 9 + \frac{14}{x} \] **Domain of \( (f + g)(x) \):** Both \( f(x) \) and \( g(x) \) are defined for all real numbers except \( x = 0 \). Therefore, the domain of \( (f + g)(x) \) is: \[ (-\infty, 0) \cup (0, \infty) \] --- ### 2. Difference of the Functions: \( (f - g)(x) \) \[ (f - g)(x) = \left(9 + \frac{7}{x}\right) - \frac{7}{x} = 9 \] **Domain of \( (f - g)(x) \):** Even though the resulting function is a constant \( 9 \), both \( f(x) \) and \( g(x) \) must be defined for \( (f - g)(x) \) to exist. Thus, the domain is: \[ (-\infty, 0) \cup (0, \infty) \] --- ### 3. Product of the Functions: \( (f \cdot g)(x) \) \[ (f \cdot g)(x) = \left(9 + \frac{7}{x}\right) \cdot \frac{7}{x} = \frac{63}{x} + \frac{49}{x^2} \] **Domain of \( (f \cdot g)(x) \):** Both \( f(x) \) and \( g(x) \) are defined for all real numbers except \( x = 0 \). Therefore, the domain of \( (f \cdot g)(x) \) is: \[ (-\infty, 0) \cup (0, \infty) \] --- ### Summary \[ \begin{align*} (f + g)(x) &= 9 + \frac{14}{x} \\ \text{Domain of } (f + g)(x) &: (-\infty, 0) \cup (0, \infty) \\ (f - g)(x) &= 9 \\ \text{Domain of } (f - g)(x) &: (-\infty, 0) \cup (0, \infty) \\ (f \cdot g)(x) &= \frac{63}{x} + \frac{49}{x^2} \\ \text{Domain of } (f \cdot g)(x) &: (-\infty, 0) \cup (0, \infty) \end{align*} \]