Let's solve this step by step.

The function $g(x)$ is given as: $g(x) = \frac{2\sqrt{x}}{3} + 7$

We need to find the inverse function $g^{-1}(x)$.

Steps to Find $g^{-1}(x)$:

1. Replace $g(x)$ with $y$: $y = \frac{2\sqrt{x}}{3} + 7$

2. Swap $x$ and $y$ (to reflect the inverse relationship): $x = \frac{2\sqrt{y}}{3} + 7$

3. Solve for $y$:

   • Subtract 7 from both sides: $x - 7 = \frac{2\sqrt{y}}{3}$

   • Multiply through by 3: $3(x - 7) = 2\sqrt{y}$

   • Divide by 2: $\frac{3(x - 7)}{2} = \sqrt{y}$

   • Square both sides to remove the square root: $y = \left( \frac{3(x - 7)}{2} \right)^2$

Thus, the inverse function is: $g^{-1}(x) = \left( \frac{3(x - 7)}{2} \right)^2$

Let me know if you'd like further explanation or assistance! Here are some additional questions and tips related to this:

Questions:

1. How do you verify if a function and its inverse are correct?
2. Can the domain of $g(x)$ affect the inverse function?
3. What does the graph of $g(x)$ and $g^{-1}(x)$ look like?
4. How does the square root in $g(x)$ impact the range of $g^{-1}(x)$?
5. Can you find $g^{-1}(x)$ for a similar function like $g(x) = \frac{\sqrt{x} + 2}{4}$?

Tip:

Always check if the function and its inverse satisfy $g(g^{-1}(x)) = x$ and $g^{-1}(g(x)) = x$ to confirm your solution