To find the inverse of the function $g(x) = -2 \sqrt{\frac{1}{3}(x + 1)} + 4$, follow these steps:

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

This step is to help with the algebra. $y = -2 \sqrt{\frac{1}{3}(x + 1)} + 4$

Step 2: Swap $x$ and $y$

To find the inverse function, interchange $x$ and $y$. $x = -2 \sqrt{\frac{1}{3}(y + 1)} + 4$

Step 3: Isolate the square root term

Subtract 4 from both sides: $x - 4 = -2 \sqrt{\frac{1}{3}(y + 1)}$

Divide both sides by -2: $\frac{x - 4}{-2} = \sqrt{\frac{1}{3}(y + 1)}$

Simplify the left side: $\frac{4 - x}{2} = \sqrt{\frac{1}{3}(y + 1)}$

Step 4: Eliminate the square root

Square both sides to remove the square root: $\left(\frac{4 - x}{2}\right)^2 = \frac{1}{3}(y + 1)$

Expand the left side: $\frac{(4 - x)^2}{4} = \frac{1}{3}(y + 1)$

Step 5: Solve for $y$

Multiply both sides by 3 to eliminate the fraction on the right: $\frac{3(4 - x)^2}{4} = y + 1$

Subtract 1 from both sides: $y = \frac{3(4 - x)^2}{4} - 1$

Step 6: Write the inverse function

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

This is the inverse of the given function.

Do you want details on any of these steps or have any questions?

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Here are 5 related questions to explore:

1. How do you check if two functions are inverses of each other?
2. What are the domain and range of the inverse function $g^{-1}(x)$?
3. Can you find the derivative of the inverse function $g^{-1}(x)$?
4. How do transformations (shifts, stretches) affect the inverse of a function?
5. What is the significance of the inverse function in real-world applications?

Tip: Always verify if the inverse function works by substituting it back into the original equation.