\( 1 \leftarrow \) For \( f(x)=\frac{x}{x-4} \) and \( g(x)=-\frac{9}{x} \), find the following composite functions and state the domain of each. \( \begin{array}{llll}\text { (a) } f \circ g & \text { (b) } g \circ f & \text { (c) } f \circ f & \text { (d) } g \circ g\end{array} \) (a) \( (f \circ g)(x)=\square \) (Simplify your answer. Use integers or fractions for any numbers in the expression.)

To find the composite functions \( f \circ g \), \( g \circ f \), \( f \circ f \), and \( g \circ g \), we will follow these steps: 1. **Calculate each composite function.** 2. **Determine the domain of each composite function.** ### Given Functions: - \( f(x) = \frac{x}{x-4} \) - \( g(x) = -\frac{9}{x} \) ### (a) Find \( (f \circ g)(x) \) This means we need to substitute \( g(x) \) into \( f(x) \): \[ (f \circ g)(x) = f(g(x)) = f\left(-\frac{9}{x}\right) \] Now, substitute \( -\frac{9}{x} \) into \( f(x) \): \[ f\left(-\frac{9}{x}\right) = \frac{-\frac{9}{x}}{-\frac{9}{x} - 4} \] Now simplify the denominator: \[ -\frac{9}{x} - 4 = -\frac{9}{x} - \frac{4x}{x} = -\frac{9 + 4x}{x} \] Thus, we have: \[ f\left(-\frac{9}{x}\right) = \frac{-\frac{9}{x}}{-\frac{9 + 4x}{x}} = \frac{-9}{-(9 + 4x)} = \frac{9}{9 + 4x} \] So, \[ (f \circ g)(x) = \frac{9}{9 + 4x} \] ### Domain of \( (f \circ g)(x) \) The domain of \( g(x) \) is \( x \neq 0 \) (since \( g(x) = -\frac{9}{x} \) is undefined at \( x = 0 \)). Next, we need to ensure that the output of \( g(x) \) does not make \( f(x) \) undefined. \( f(x) \) is undefined when \( x - 4 = 0 \) or \( x = 4 \). Set \( g(x) = -\frac{9}{x} = 4 \): \[ -\frac{9}{x} = 4 \implies -9 = 4x \implies x = -\frac{9}{4} \] Thus, the domain of \( (f \circ g)(x) \) is: \[ x \neq 0 \quad \text{and} \quad x \neq -\frac{9}{4} \] ### Final Answer for (a): \[ (f \circ g)(x) = \frac{9}{9 + 4x} \] Domain: \( x \neq 0, x \neq -\frac{9}{4} \) Now, let's proceed to calculate the other composite functions. ### (b) Find \( (g \circ f)(x) \) This means we need to substitute \( f(x) \) into \( g(x) \): \[ (g \circ f)(x) = g(f(x)) = g\left(\frac{x}{x-4}\right) \] Substituting into \( g(x) \): \[ g\left(\frac{x}{x-4}\right) = -\frac{9}{\frac{x}{x-4}} = -\frac{9(x-4)}{x} = -\frac{9x - 36}{x} = -9 + \frac{36}{x} \] ### Domain of \( (g \circ f)(x) \) The domain of \( f(x) \) is \( x \neq 4 \). Next, we need to ensure that the output of \( f(x) \) does not make \( g(x) \) undefined. \( g(x) \) is undefined when \( x = 0 \). Set \( f(x) = \frac{x}{x-4} = 0 \): \[ \frac{x}{x-4} = 0 \implies x = 0 \] Thus, the domain of \( (g \circ f)(x) \) is: \[ x \neq 4 \quad \text{and} \quad x \neq 0 \] ### Final Answer for (b): \[ (g \circ f)(x) = -9 + \frac{36}{x} \] Domain: \( x \neq 4, x \neq 0 \) ### (c) Find \( (f \circ f)(x) \) This means we need to substitute \( f(x) \) into itself: \[ (f \circ f)(x) = f(f(x)) = f\left(\frac{x}{x-4}\right) \] Substituting into \( f(x) \): \[ f\left(\frac{x}{x-4}\right) = \frac{\frac{x}{x-4}}{\frac{x}{x-4} - 4} \] Now simplify the denominator: \[ \frac{x}{x-4} - 4 = \frac{x - 4(x-4)}{x-4} = \frac{x - 4x + 16}{x-4} = \frac{-3x + 16}{x-4} \] Thus, we have: \[ f\left(\frac{x}{x-4}\right) = \frac{\frac{x}{x-4}}{\frac{-3x + 16}{x-4}} = \frac{x}{-3x + 16} \] ### Domain of \( (f \circ f)(x) \) The domain of \( f(x) \) is \( x \neq 4 \). Next, we need to ensure that the output of \( f(x) \) does not make \( f(x) \) undefined again. Set \( f(x) = \frac{x}{x-4} = 4 \): \[ \frac{x}{x-4} = 4 \implies x = 4(x-4) \implies x = 4x - 16 \implies 3x = 16 \implies x = \frac{16}{3} \] Thus, the domain of \( (f \circ f)(x) \) is: \[ x \neq 4 \quad \text{and} \quad x \neq \frac{16}{3} \] ### Final Answer for (c): \[ (f \circ f)(x) = \frac{x}{-3x + 16} \] Domain: \( x \neq 4, x \neq \frac{16}{3} \) ### (d) Find \( (g \circ g)(x) \) This means we need to substitute \( g(x) \)