Let's find the inverses of each of the given functions step by step.

1. $f(x) = x^3 - 15$

To find the inverse, follow these steps:

1. Replace $f(x)$ with $y$:
   $y = x^3 - 15$
2. Swap $x$ and $y$:
   $x = y^3 - 15$
3. Solve for $y$:
   $x + 15 = y^3$
   $y = \sqrt[3]{x + 15}$

Thus, the inverse of $f(x)$ is:
$f^{-1}(x) = \sqrt[3]{x + 15}$

2. $g(x) = \frac{2x}{3}$

1. Replace $g(x)$ with $y$:
   $y = \frac{2x}{3}$
2. Swap $x$ and $y$:
   $x = \frac{2y}{3}$
3. Solve for $y$:
   $y = \frac{3x}{2}$

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

3. $h(x) = \frac{x^2}{5}$

1. Replace $h(x)$ with $y$:
   $y = \frac{x^2}{5}$
2. Swap $x$ and $y$:
   $x = \frac{y^2}{5}$
3. Solve for $y$:
   $y = \pm \sqrt{5x}$

Thus, the inverse of $h(x)$ is:
$h^{-1}(x) = \pm \sqrt{5x}$

4. $2y = 3x$

1. Solve for $y$:
   $y = \frac{3x}{2}$

This is already the equation for $y$, and thus the inverse is the same.
$y = \frac{3x}{2}$

5. $i(x) = \frac{x - 5}{3}$

1. Replace $i(x)$ with $y$:
   $y = \frac{x - 5}{3}$
2. Swap $x$ and $y$:
   $x = \frac{y - 5}{3}$
3. Solve for $y$:
   $y - 5 = 3x$
   $y = 3x + 5$

Thus, the inverse of $i(x)$ is:
$i^{-1}(x) = 3x + 5$

Do you need further details or have any questions?

Here are 5 related questions:

1. How can we verify if a function and its inverse are correct?
2. What does it mean for a function to be one-to-one, and why is this important when finding inverses?
3. Can all functions have inverses? If not, which ones cannot?
4. How do graphical representations of a function and its inverse relate to each other?
5. How can we find the inverse of more complicated functions like logarithmic or exponential functions?

Tip: The inverse of a function can be thought of as "undoing" the effect of the original function. It reverses the inputs and outputs.