To find the inverse of the given function:

$g(x) = \frac{-4 + \sqrt[3]{4x}}{2}$

we follow these steps:

Step 1: Replace $g(x)$ with $y$.

$y = \frac{-4 + \sqrt[3]{4x}}{2}$

Step 2: Swap $x$ and $y$ to find the inverse.

$x = \frac{-4 + \sqrt[3]{4y}}{2}$

Step 3: Solve for $y$.

1. Multiply both sides by 2 to get rid of the denominator: $2x = -4 + \sqrt[3]{4y}$

2. Add 4 to both sides: $2x + 4 = \sqrt[3]{4y}$

3. Cube both sides to eliminate the cube root: $(2x + 4)^3 = 4y$

4. Divide both sides by 4: $y = \frac{(2x + 4)^3}{4}$

Step 4: Write the inverse function.

The inverse of $g(x)$ is: $g^{-1}(x) = \frac{(2x + 4)^3}{4}$

Would you like any further details on these steps or have any questions?

Here are some related questions to expand on this topic:

1. What is the significance of finding the inverse of a function in real-world applications?
2. How do you verify if two functions are truly inverses of each other?
3. Can all functions have inverses? If not, under what conditions do functions lack inverses?
4. How do transformations (like translations or dilations) affect the inverse of a function?
5. What are the graphical interpretations of a function and its inverse?

Tip: Always check if the original function is one-to-one before finding its inverse, as only one-to-one functions have valid inverses.