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

To find the inverse function \( f^{-1}(x) \) for the given function \( f(x) = \frac{4}{x + 2} + 1 \), let's follow these steps in a fun way: 1. **Switch the roles of \( x \) and \( y \)**: Start by rewriting the function as \( y = \frac{4}{x + 2} + 1 \). Now, exchange \( x \) and \( y \) to get \( x = \frac{4}{y + 2} + 1 \). 2. **Isolate \( y \)**: Next, we need to solve for \( y \). First, subtract 1 from both sides: \[ x - 1 = \frac{4}{y + 2} \] 3. **Multiply both sides by \( (y + 2) \)**: This gives us: \[ (x - 1)(y + 2) = 4 \] 4. **Distribute**: This expands to: \[ (x - 1)y + 2(x - 1) = 4 \] 5. **Rearranging the equation**: We rewrite it as: \[ (x - 1)y = 4 - 2(x - 1) \] which simplifies to: \[ (x - 1)y = 4 - 2x + 2 \] and further to: \[ (x - 1)y = 6 - 2x \] 6. **Solve for \( y \)**: Finally, divide both sides by \( (x - 1) \): \[ y = \frac{6 - 2x}{x - 1} \] Thus, the inverse function is: \[ f^{-1}(x) = \frac{6 - 2x}{x - 1} \] Isn’t that nifty? You’ve just unlocked the mysteries of inverses! Give yourself a high five! 🎉