We are tasked with finding the inverse of the function $y = 3x - 8$ and determining which of the options (A, B, C, or D) is equivalent to this inverse.

Steps to find the inverse:

1. Start with the equation: $y = 3x - 8$
2. Swap $x$ and $y$ (this is the process for finding the inverse function): $x = 3y - 8$
3. Solve for $y$: $x + 8 = 3y$ $y = \frac{x + 8}{3}$ This is the inverse function.

Now, let's examine the choices:

• A: $y = \frac{1}{3x - 8}$ — This is not equivalent to the inverse function.
• B: $y = \frac{3}{x} - 8$ — Also not equivalent.
• C: $-y = -3x + 8$ — Simplifying this: $y = 3x - 8$, which is just the original function, not the inverse.
• D: $x = 3y - 8$ — This matches the equation we had right before solving for $y$, so this is equivalent to the inverse.

Correct Answer: D

Would you like further clarification or more details on this problem?

Here are some related questions:

1. How do you verify an inverse function?
2. What is the significance of swapping $x$ and $y$ when finding an inverse?
3. Can a function always have an inverse?
4. How do horizontal line tests relate to inverse functions?
5. How do you graph the inverse of a function?

Tip: To check if two functions are inverses, you can compose them: if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$, then they are inverses.