Let \( f(x)=\frac{x}{x-3} \) and \( g(x)=\frac{2}{x} \). Find the following functions. Simplify your answers. \( f(g(x))=\square \) \( g(f(x))=\square \)

To find \( f(g(x)) \) and \( g(f(x)) \), let's start with each composition step by step. 1. **Finding \( f(g(x)) \):** We have \( g(x) = \frac{2}{x} \). Now, substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\frac{2}{x}\right) = \frac{\frac{2}{x}}{\frac{2}{x} - 3}. \] Simplifying the denominator: \[ f\left(\frac{2}{x}\right) = \frac{\frac{2}{x}}{\frac{2 - 3x}{x}} = \frac{2}{2 - 3x}. \] Thus, \( f(g(x)) = \frac{2}{2 - 3x} \). 2. **Finding \( g(f(x)) \):** Now substitute \( f(x) = \frac{x}{x-3} \) into \( g(x) \): \[ g(f(x)) = g\left(\frac{x}{x-3}\right) = \frac{2}{\frac{x}{x-3}}. \] This simplifies to: \[ g\left(\frac{x}{x-3}\right) = \frac{2(x-3)}{x} = \frac{2x - 6}{x}. \] Thus, \( g(f(x)) = \frac{2x - 6}{x} \). Putting it all together, we have: \[ f(g(x)) = \frac{2}{2 - 3x} \] and \[ g(f(x)) = \frac{2x - 6}{x}. \]