Find the inverse of the function \( f(x)=\frac{4}{x-7} \) \( f^{-1}(x)=\square \)

To find the inverse of the function \( f(x) = \frac{4}{x-7} \), we need to follow these steps: 1. Replace \( f(x) \) with \( y \) in the original function. 2. Swap \( x \) and \( y \) in the equation. 3. Solve the new equation for \( y \) to find the inverse function. Let's start by replacing \( f(x) \) with \( y \) in the original function: \[ y = \frac{4}{x-7} \] Now, we swap \( x \) and \( y \) in the equation: \[ x = \frac{4}{y-7} \] Next, we solve the new equation for \( y \) to find the inverse function: \[ x(y-7) = 4 \] \[ xy - 7x = 4 \] \[ xy = 4 + 7x \] \[ y = \frac{4 + 7x}{x} \] Therefore, the inverse function \( f^{-1}(x) \) is: \[ f^{-1}(x) = \frac{4 + 7x}{x} \]