Which function is the inverse of \( g(x)=2 \sqrt[3]{x-3}+4 \) ? \( f(x)=\frac{(x-4)^{2}}{8}+3 \) \( f(x)=\frac{(x-2)^{2}}{64}+3 \) \( f(x)=\frac{(x-4)^{3}}{2}+3 \) \( f(x)=\frac{(x-2)^{3}}{4}+3 \)

To find the inverse of the function \( g(x) = 2 \sqrt[3]{x-3} + 4 \), we need to switch the roles of \( x \) and \( y \) and solve for \( y \). Let's denote \( y = g(x) \) and then solve for \( x \): \[ y = 2 \sqrt[3]{x-3} + 4 \] First, we'll isolate the cube root term by subtracting 4 from both sides: \[ y - 4 = 2 \sqrt[3]{x-3} \] Next, we'll divide both sides by 2 to get the cube root term by itself: \[ \frac{y - 4}{2} = \sqrt[3]{x-3} \] Now, we'll cube both sides to eliminate the cube root: \[ \left(\frac{y - 4}{2}\right)^3 = x - 3 \] Add 3 to both sides to isolate \( x \): \[ \left(\frac{y - 4}{2}\right)^3 + 3 = x \] Now, we'll solve for \( y \) by switching \( x \) and \( y \): \[ y = \left(\frac{x - 4}{2}\right)^3 + 3 \] This is the inverse function of \( g(x) \). Let's compare this with the given options: 1. \( f(x) = \frac{(x-4)^2}{8} + 3 \) 2. \( f(x) = \frac{(x-2)^2}{64} + 3 \) 3. \( f(x) = \frac{(x-4)^3}{2} + 3 \) 4. \( f(x) = \frac{(x-2)^3}{4} + 3 \) The correct inverse function is the one that matches the form we derived: \[ y = \left(\frac{x - 4}{2}\right)^3 + 3 \] This matches option 3: \[ f(x) = \frac{(x-4)^3}{2} + 3 \] Therefore, the correct answer is: \( f(x) = \frac{(x-4)^3}{2} + 3 \)