Let \( f(x)=\frac{1}{x-5} \) and \( g(x)=\frac{2}{x}+5 \) Find the following functions. Simplify your answers. \( f(g(x))=\square \) \( g(f(x))=\square \)

To find the composite functions \( f(g(x)) \) and \( g(f(x)) \), we will substitute one function into the other. 1. **Finding \( f(g(x)) \)**: \[ g(x) = \frac{2}{x} + 5 \] Now substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\frac{2}{x} + 5\right) = \frac{1}{\left(\frac{2}{x} + 5\right) - 5} \] Simplifying the expression inside the function: \[ \frac{2}{x} + 5 - 5 = \frac{2}{x} \] Thus, \[ f(g(x)) = \frac{1}{\frac{2}{x}} = \frac{x}{2} \] 2. **Finding \( g(f(x)) \)**: \[ f(x) = \frac{1}{x-5} \] Now substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g\left(\frac{1}{x-5}\right) = \frac{2}{\frac{1}{x-5}} + 5 \] Simplifying: \[ \frac{2}{\frac{1}{x-5}} = 2(x-5) = 2x - 10 \] So, \[ g(f(x)) = 2x - 10 + 5 = 2x - 5 \] Final answers: \[ f(g(x)) = \frac{x}{2} \] \[ g(f(x)) = 2x - 5 \]