Find \( f^{-1}(x) \) iff \( f(x)=\frac{3 x+4}{x+1} \)

To find the inverse function \( f^{-1}(x) \) of \( f(x) = \frac{3x + 4}{x + 1} \), we start by replacing \( f(x) \) with \( y \): \[ y = \frac{3x + 4}{x + 1} \] Next, we need to solve for \( x \) in terms of \( y \). We multiply both sides by \( x + 1 \) to eliminate the fraction: \[ y(x + 1) = 3x + 4 \] Expanding the left side gives us: \[ yx + y = 3x + 4 \] Now, we want to isolate \( x \). We can rearrange the equation: \[ yx - 3x = 4 - y \] Factoring \( x \) out on the left side yields: \[ x(y - 3) = 4 - y \] Now, dividing both sides by \( y - 3 \), we have: \[ x = \frac{4 - y}{y - 3} \] Since we originally set \( y = f(x) \), we can now replace \( y \) with \( x \) to get \( f^{-1}(x) \): \[ f^{-1}(x) = \frac{4 - x}{x - 3} \] So, the inverse function is: \[ f^{-1}(x) = \frac{4 - x}{x - 3} \]