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 \( f(g(x)) \): First, substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\frac{2}{x}\right) = \frac{\frac{2}{x}}{\frac{2}{x} - 3} \] Now simplify the denominator: \[ \frac{2}{x} - 3 = \frac{2 - 3x}{x} \] So we have: \[ f(g(x)) = \frac{\frac{2}{x}}{\frac{2 - 3x}{x}} = \frac{2}{2 - 3x} \] Next, to find \( g(f(x)) \): Substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g\left(\frac{x}{x-3}\right) = \frac{2}{\frac{x}{x-3}} = \frac{2(x-3)}{x} \] Now simplify: \[ g(f(x)) = \frac{2x - 6}{x} \] Thus, we have simplified the functions to: \[ f(g(x)) = \frac{2}{2 - 3x} \quad \text{and} \quad g(f(x)) = \frac{2x - 6}{x} \]