To find the inverse of the function $g(x) = \frac{4x+6}{5}$, we follow these steps:

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

$y = \frac{4x+6}{5}$

Step 2: Swap $x$ and $y$ (to solve for the inverse)

$x = \frac{4y+6}{5}$

Step 3: Solve for $y$

Multiply both sides by 5 to eliminate the denominator: $5x = 4y + 6$

Subtract 6 from both sides: $5x - 6 = 4y$

Divide by 4: $y = \frac{5x - 6}{4}$

Thus, the inverse function is: $g^{-1}(x) = \frac{5x - 6}{4}$

Step 4: Match the result to the answer choices

The correct answer is D: $\frac{5x - 6}{4}$.

Do you want a detailed breakdown of each calculation? Here are some related questions to expand understanding:

1. What is the general method for finding inverse functions?
2. How do you verify that two functions are inverses of each other?
3. What is the domain of the inverse function in this example?
4. How do restrictions on the original function affect its inverse?
5. Can every function have an inverse? Why or why not?

Tip: When finding inverses, always check by substituting the result back into the original function to ensure they cancel out.