Problem
Find the inverse of the one-to-one function. \[ f(x)=-3 x-6 \] \[ f^{-1}(x)= \] (A) \[ \frac{x-6}{-3} \] \[ f^{11}(x)= \] (B) \[ \frac{-3 x+6}{-3} \] \[ f^{7}(x)= \] (C) \[ \frac{y+6}{-3} \] $f^{\prime}(x)=$ (D) \[ \frac{x+6}{-3} \]
Solution
Step 1 ：Given function: \(f(x) = -3x - 6\)
Step 2 ：Replace \(f(x)\) with y: \(y = -3x - 6\)
Step 3 ：Swap x and y: \(x = -3y - 6\)
Step 4 ：Solve for y: \(y = \frac{x+6}{-3}\)
Step 5 ：\(\boxed{f^{-1}(x) = \frac{x+6}{-3}}\)
