Part B b. \( g(x)=\frac{1}{3} x-12 \) \[ \text { A) } g^{-1}(x)=-\frac{1}{3} x+12 \] B) \( g^{-1}(x)=-3 x+\frac{1}{12} \) C) \( g^{-1}(x)=3 x+36 \) D) \( g^{-1}(x)=-12 x+\frac{1}{3} \)

To find the inverse of the function \( g(x) = \frac{1}{3}x - 12 \), we start by swapping \( g(x) \) with \( x \) and then solving for \( g^{-1}(x) \). 1. Set \( y = g(x) \): \[ y = \frac{1}{3}x - 12 \] 2. Swap \( y \) and \( x \): \[ x = \frac{1}{3}y - 12 \] 3. Solve for \( y \): \[ x + 12 = \frac{1}{3}y \] \[ y = 3(x + 12) \] \[ y = 3x + 36 \] Thus, the inverse function is \( g^{-1}(x) = 3x + 36 \), which corresponds to option C. --- Finding inverse functions can be a bit tricky if you're not careful! A common mistake is forgetting to swap \( x \) and \( y \)—always remember that this step is crucial for determining the correct inverse relationship. Another tip: once you have the expression for the inverse, plug in a value from the original function to ensure it works both ways. --- If you're eager to dive deeper, there are numerous resources available! Books like "Algebra and Trigonometry" by Michael Sullivan break down functions and inverses in a fun and engaging way. Websites like Khan Academy offer great practice problems and interactive lessons that can make the process more enjoyable. Happy learning!