Find \( f \circ g \) and \( g \circ f \). \[ f(x)=\frac{1}{x}, g(x)=x+6 \] (a) \( f \circ g \) (b) \( g \circ f \) Find the domain of each function and each composite function. (Enter your answers using interval notation.) domain of \( f \) domain of \( g \) domain of \( f \circ g \) domain of \( g \circ f \)

To find \( f \circ g \) and \( g \circ f \), we start with the definitions of the functions: 1. \( f(x) = \frac{1}{x} \) 2. \( g(x) = x + 6 \) (a) **Finding \( f \circ g \)**: \[ f \circ g(x) = f(g(x)) = f(x + 6) = \frac{1}{x + 6} \] (b) **Finding \( g \circ f \)**: \[ g \circ f(x) = g(f(x)) = g\left(\frac{1}{x}\right) = \frac{1}{x} + 6 \] Now, we will determine the domain of each function and the composite functions. 1. **Domain of \( f \)**: - \( f(x) = \frac{1}{x} \) is defined for \( x \neq 0 \). - Therefore, the domain is \( (-\infty, 0) \cup (0, \infty) \). 2. **Domain of \( g \)**: - \( g(x) = x + 6 \) is defined for all \( x \). - Hence, the domain is \( (-\infty, \infty) \). 3. **Domain of \( f \circ g \)**: - Since \( g(x) = x + 6 \) is defined for all \( x \) and \( f(x) \) is only undefined at \( x = -6 \) (where \( g(x) = -6 \)): - The domain of \( f \circ g \) is \( (-\infty, -6) \cup (-6, \infty) \). 4. **Domain of \( g \circ f \)**: - Since \( f(x) = \frac{1}{x} \) is defined for \( x \neq 0 \) and \( g(x) \) is defined for all values: - The domain of \( g \circ f \) is \( (-\infty, 0) \cup (0, \infty) \). Here’s a summary of the domains: - Domain of \( f \): \( (-\infty, 0) \cup (0, \infty) \) - Domain of \( g \): \( (-\infty, \infty) \) - Domain of \( f \circ g \): \( (-\infty, -6) \cup (-6, \infty) \) - Domain of \( g \circ f \): \( (-\infty, 0) \cup (0, \infty) \)