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

To find the inverse of the function \( f(x) = -3x - 12 \), we start by swapping \( f(x) \) with \( y \): 1. Set \( y = -3x - 12 \). 2. Now, solve for \( x \): \[ y + 12 = -3x \implies x = -\frac{1}{3}(y + 12) \] Simplifying gives: \[ x = -\frac{1}{3}y - 4 \] 3. Now, replace \( y \) with \( x \) to express the inverse function: \[ f^{-1}(x) = -\frac{1}{3}x - 4 \] So the correct answer is option B) \( f^{-1}(x) = -\frac{1}{3}x - 4 \). To verify this inverse, we can check if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) holds true: For \( f(f^{-1}(x)) \): 1. Substitute \( f^{-1}(x) \) into \( f \): \[ f(f^{-1}(x)) = f\left(-\frac{1}{3}x - 4\right) = -3\left(-\frac{1}{3}x - 4\right) - 12 = x + 12 - 12 = x \] For \( f^{-1}(f(x)) \): 1. Substitute \( f(x) \) into \( f^{-1} \): \[ f^{-1}(f(x)) = f^{-1}(-3x - 12) = -\frac{1}{3}(-3x - 12) - 4 = x + 4 - 4 = x \] Thus confirming that the inverse function is correctly derived!