To find the inverse of the function(s) given, we need to:

1. Swap $x$ and $y$.
2. Solve for $y$.

Given two equations:

1. $f(x) = \frac{f(x) + 3}{x + \frac{5}{x} - 2},$ $x \neq 22$
2. $f(x) = \frac{4x + 3}{6 - 5x},$ $x \neq \frac{6}{5}$

However, the first function appears to be incorrectly formatted. Let’s address the second equation first:

For $f(x) = \frac{4x + 3}{6 - 5x}$:

1. Swap $f(x)$ with $x$ and $x$ with $y$: $x = \frac{4y + 3}{6 - 5y}$

2. Solve for $y$: $x(6 - 5y) = 4y + 3$ $6x - 5xy = 4y + 3$ $6x - 3 = 4y + 5xy$ $6x - 3 = y(4 + 5x)$ $y = \frac{6x - 3}{4 + 5x}$

So, the inverse function is: $f^{-1}(x) = \frac{6x - 3}{4 + 5x}$

For the first function:

Since the equation seems incorrect or unclear, I need the correct function format to provide the inverse.

Do you want details on this, or have any questions?

Here are five related questions:

1. Can you confirm the first function, or would you like to correct it?
2. How would you verify that the functions are inverses of each other?
3. What are the conditions under which the inverses do not exist?
4. Can you plot the original and inverse functions to visualize them?
5. How does the domain and range of the original function relate to the inverse function?

Tip: When finding the inverse, always ensure the function is one-to-one, meaning it passes the horizontal line test.