Find \( f^{-1}(x) \) for the following function: \( f(x)=\frac{3}{x+2}+4 \) \( f^{-1}(x)= \)

To find \( f^{-1}(x) \), we start by setting \( y = f(x) \), which gives us: \[ y = \frac{3}{x+2} + 4. \] Next, we solve for \( x \) in terms of \( y \): 1. Subtract 4 from both sides: \[ y - 4 = \frac{3}{x+2}. \] 2. Multiply both sides by \( x + 2 \): \[ (y - 4)(x + 2) = 3. \] 3. Expand and rearrange: \[ yx - 4x + 2y - 8 = 3 \] \[ yx - 4x = 3 - 2y + 8 \] \[ x(y - 4) = 11 - 2y. \] 4. Finally, divide both sides by \( y - 4 \): \[ x = \frac{11 - 2y}{y - 4}. \] Thus, replacing \( y \) with \( x \) to express the inverse function gives us: \[ f^{-1}(x) = \frac{11 - 2x}{x - 4}. \]