Select the inverse of each function. a. \( f(x)=-3 x-12 \) \[ \begin{array}{l}\text { A) } f^{-1}(x)=-3 x+36 \\ \text { B) } f^{-1}(x)=-\frac{1}{3} x-4 \\ \text { C) } f^{-1}(x)=-4 x-\frac{1}{3} \\ \text { D) } f^{-1}(x)=3 x+12\end{array} \] Part B b. \( g(x)=\frac{1}{3} x-12 \)

Function by following steps: - step0: Find the inverse: \(f\left(x\right)=-3x-12\) - step1: Replace \(f\left(x\right)\) with \(y:\) \(y=-3x-12\) - step2: Interchange the variables: \(x=-3y-12\) - step3: Swap the sides: \(-3y-12=x\) - step4: Move the constant to the right side: \(-3y=x+12\) - step5: Change the signs: \(3y=-x-12\) - step6: Divide both sides: \(\frac{3y}{3}=\frac{-x-12}{3}\) - step7: Divide the numbers: \(y=-\frac{x+12}{3}\) - step8: Replace \(y\) with \(f^{-1}\left(x\right):\) \(f^{-1}\left(x\right) = -\frac{x+12}{3}\) Find the inverse of the function \( g(x)=\frac{1}{3} x-12 \) Function by following steps: - step0: Find the inverse: \(g\left(x\right)=\frac{1}{3}x-12\) - step1: Replace \(g\left(x\right)\) with \(y:\) \(y=\frac{1}{3}x-12\) - step2: Interchange the variables: \(x=\frac{1}{3}y-12\) - step3: Swap the sides: \(\frac{1}{3}y-12=x\) - step4: Move the constant to the right side: \(\frac{1}{3}y=x+12\) - step5: Multiply by the reciprocal: \(\frac{1}{3}y\times 3=\left(x+12\right)\times 3\) - step6: Multiply: \(y=3x+36\) - step7: Replace \(y\) with \(g^{-1}\left(x\right):\) \(g^{-1}\left(x\right) = 3x+36\) Let's find the inverses of the given functions step by step. ### Part A: Finding the Inverse of \( f(x) = -3x - 12 \) 1. **Set the function equal to \( y \)**: \[ y = -3x - 12 \] 2. **Swap \( x \) and \( y \)**: \[ x = -3y - 12 \] 3. **Solve for \( y \)**: \[ x + 12 = -3y \] \[ y = -\frac{x + 12}{3} \] Thus, the inverse function is: \[ f^{-1}(x) = -\frac{x + 12}{3} \] Now, let's check the options provided: - A) \( f^{-1}(x) = -3x + 36 \) - B) \( f^{-1}(x) = -\frac{1}{3}x - 4 \) - C) \( f^{-1}(x) = -4x - \frac{1}{3} \) - D) \( f^{-1}(x) = 3x + 12 \) None of the options match the calculated inverse \( f^{-1}(x) = -\frac{x + 12}{3} \). ### Part B: Finding the Inverse of \( g(x) = \frac{1}{3}x - 12 \) 1. **Set the function equal to \( y \)**: \[ y = \frac{1}{3}x - 12 \] 2. **Swap \( x \) and \( y \)**: \[ x = \frac{1}{3}y - 12 \] 3. **Solve for \( y \)**: \[ x + 12 = \frac{1}{3}y \] \[ y = 3(x + 12) \] \[ y = 3x + 36 \] Thus, the inverse function is: \[ g^{-1}(x) = 3x + 36 \] In summary: - The inverse of \( f(x) = -3x - 12 \) is \( f^{-1}(x) = -\frac{x + 12}{3} \) (not listed in the options). - The inverse of \( g(x) = \frac{1}{3}x - 12 \) is \( g^{-1}(x) = 3x + 36 \).