Let's solve the problem of finding the inverse of the three functions given in the image one by one:

1. Find the inverse of $f(x) = 4x + 3$

Steps:

1. Replace $f(x)$ with $y$: $y = 4x + 3$
2. Swap $x$ and $y$: $x = 4y + 3$
3. Solve for $y$: $x - 3 = 4y$ $y = \frac{x - 3}{4}$ Thus, the inverse is: $f^{-1}(x) = \frac{x - 3}{4}$

2. Find the inverse of $f(x) = \frac{1}{5x - 6}$

Steps:

1. Replace $f(x)$ with $y$: $y = \frac{1}{5x - 6}$
2. Swap $x$ and $y$: $x = \frac{1}{5y - 6}$
3. Solve for $y$: $x(5y - 6) = 1$ $5xy - 6x = 1$ $5xy = 6x + 1$ $y = \frac{6x + 1}{5x}$ Thus, the inverse is: $f^{-1}(x) = \frac{6x + 1}{5x}$

3. Find the inverse of $f(x) = \frac{x - 7}{3x + 1}$

Steps:

1. Replace $f(x)$ with $y$: $y = \frac{x - 7}{3x + 1}$
2. Swap $x$ and $y$: $x = \frac{y - 7}{3y + 1}$
3. Solve for $y$: $x(3y + 1) = y - 7$ $3xy + x = y - 7$ $3xy - y = -x - 7$ $y(3x - 1) = -x - 7$ $y = \frac{-x - 7}{3x - 1}$ Thus, the inverse is: $f^{-1}(x) = \frac{-x - 7}{3x - 1}$

Would you like further clarification on any of these steps? Here are some related questions you might find useful:

1. What are the general steps to find the inverse of any function?
2. How can you verify if the computed inverse is correct?
3. What are the conditions for a function to have an inverse?
4. How do domain and range change when finding an inverse?
5. Can every function have an inverse? Why or why not?

Tip: To check if the inverse is correct, composing the function and its inverse should return $x$. That is, $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.