Let's solve this inverse function matching problem by analyzing each pair.

We need to match each function with its inverse function. The given functions are:

1. $f(x) = \frac{2x}{3} - 17$
2. $f(x) = x - 10$
3. $f(x) = \sqrt[3]{2x}$
4. $f(x) = \frac{x}{5}$

And the possible inverse functions are:

• $f^{-1}(x) = 5x$
• $f^{-1}(x) = \frac{3x}{2} + 17$
• $f^{-1}(x) = x + 10$
• $f^{-1}(x) = \frac{x^3}{2}$

Steps to match each function with its inverse:

1. For $f(x) = \frac{2x}{3} - 17$:

   • To find the inverse, we solve $y = \frac{2x}{3} - 17$ for $x$.
   • $y + 17 = \frac{2x}{3}$
   • Multiply both sides by 3: $3(y + 17) = 2x$
   • Divide by 2: $x = \frac{3(y + 17)}{2}$
   • So, the inverse is $f^{-1}(x) = \frac{3(x + 17)}{2}$.

2. For $f(x) = x - 10$:

   • To find the inverse, we solve $y = x - 10$ for $x$.
   • $x = y + 10$
   • So, the inverse is $f^{-1}(x) = x + 10$.

3. For $f(x) = \sqrt[3]{2x}$:

   • To find the inverse, we cube both sides: $y^3 = 2x$.
   • Then, divide by 2: $x = \frac{y^3}{2}$
   • So, the inverse is $f^{-1}(x) = \frac{x^3}{2}$.

4. For $f(x) = \frac{x}{5}$:

   • To find the inverse, multiply both sides by 5: $y = \frac{x}{5} \Rightarrow x = 5y$
   • So, the inverse is $f^{-1}(x) = 5x$.

Now, we can match the functions with their inverses:

• $f(x) = \frac{2x}{3} - 17$ matches with $f^{-1}(x) = \frac{3(x + 17)}{2}$
• $f(x) = x - 10$ matches with $f^{-1}(x) = x + 10$
• $f(x) = \sqrt[3]{2x}$ matches with $f^{-1}(x) = \frac{x^3}{2}$
• $f(x) = \frac{x}{5}$ matches with $f^{-1}(x) = 5x$

Let me know if you would like further details, and feel free to ask any questions.

Here are 5 relative questions to expand on the topic:

1. How do you verify if a function and its inverse are correct?
2. What are the conditions for a function to have an inverse?
3. Can all functions have an inverse? Why or why not?
4. How can you find the inverse of more complex functions like trigonometric functions?
5. What is the graphical relationship between a function and its inverse?

Tip: A function must be one-to-one (bijective) to have an inverse, meaning each input maps to exactly one output, and vice versa.