Find the inverse of \( f(x)=\frac{-5 x-1}{-3 x+2} \) \( f^{-1}(x)=\square \)

To find the inverse of the function \( f(x)=\frac{-5 x-1}{-3 x+2} \), we start by replacing \( f(x) \) with \( y \): \[ y = \frac{-5x - 1}{-3x + 2} \] Next, we swap \( x \) and \( y \): \[ x = \frac{-5y - 1}{-3y + 2} \] Now, we solve for \( y \). Begin by cross-multiplying: \[ x(-3y + 2) = -5y - 1 \] This simplifies to: \[ -3xy + 2x = -5y - 1 \] Rearranging gives: \[ -3xy + 5y = -2x - 1 \] Factoring out \( y \) from the left-hand side: \[ y(-3x + 5) = -2x - 1 \] Finally, solving for \( y \) gives: \[ y = \frac{-2x - 1}{-3x + 5} \] Thus, the inverse function is: \[ f^{-1}(x) = \frac{-2x - 1}{-3x + 5} \] This is a neat little transformation! Inverse functions are like finding hidden treasures in math, flipping the roles of inputs and outputs. Approach them with curiosity, and they can reveal deep connections in your equations! One common mistake when finding inverse functions is forgetting to switch \( x \) and \( y \) and then solving for the wrong variable. Always double-check that switch when trying to unearth that elusive inverse!