Let's solve the given problems involving inverse and composite functions.

Given functions: $f(x) = \frac{2x - 5}{3}$ $g(x) = 11 - \frac{7x}{6}$

We need to find the following:

1. $f^{-1}(x)$
2. $g^{-1}(x)$
3. $f(g(x))$
4. $g(f(x))$
5. $g^{-1}(f(x))$
6. $f(g^{-1}(x))$
7. $g(f^{-1}(x))$

1. Find $f^{-1}(x)$

To find the inverse of $f(x)$:

$y = \frac{2x - 5}{3}$ $3y = 2x - 5$ $2x = 3y + 5$ $x = \frac{3y + 5}{2}$

So,

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

2. Find $g^{-1}(x)$

To find the inverse of $g(x)$:

$y = 11 - \frac{7x}{6}$ $\frac{7x}{6} = 11 - y$ $7x = 6(11 - y)$ $7x = 66 - 6y$ $x = \frac{66 - 6y}{7}$

So,

$g^{-1}(x) = \frac{66 - 6x}{7}$

3. Find $f(g(x))$

$g(x) = 11 - \frac{7x}{6}$ $f(g(x)) = f\left(11 - \frac{7x}{6}\right)$ $= \frac{2\left(11 - \frac{7x}{6}\right) - 5}{3}$ $= \frac{2 \times 11 - 2 \times \frac{7x}{6} - 5}{3}$ $= \frac{22 - \frac{14x}{6} - 5}{3}$ $= \frac{22 - \frac{7x}{3} - 5}{3}$ $= \frac{17 - \frac{7x}{3}}{3}$ $= \frac{17 \times 3 - 7x}{3 \times 3}$ $= \frac{51 - 7x}{9}$

4. Find $g(f(x))$

$f(x) = \frac{2x - 5}{3}$ $g(f(x)) = g\left(\frac{2x - 5}{3}\right)$ $= 11 - \frac{7\left(\frac{2x - 5}{3}\right)}{6}$ $= 11 - \frac{7(2x - 5)}{18}$ $= 11 - \frac{14x - 35}{18}$ $= 11 - \frac{14x}{18} + \frac{35}{18}$ $= 11 - \frac{7x}{9} + \frac{35}{18}$ $= 11 - \frac{7x}{9} + \frac{35}{18}$ $= \frac{198 - 7x + 35}{18}$ $= \frac{233 - 7x}{18}$

5. Find $g^{-1}(f(x))$

$f(x) = \frac{2x - 5}{3}$ $g^{-1}(x) = \frac{66 - 6x}{7}$ $g^{-1}(f(x)) = g^{-1}\left(\frac{2x - 5}{3}\right)$ $= \frac{66 - 6\left(\frac{2x - 5}{3}\right)}{7}$ $= \frac{66 - \frac{12x - 30}{3}}{7}$ $= \frac{66 - 4x + 10}{7}$ $= \frac{76 - 4x}{7}$

6. Find $f(g^{-1}(x))$

$g^{-1}(x) = \frac{66 - 6x}{7}$ $f(g^{-1}(x)) = f\left(\frac{66 - 6x}{7}\right)$ $= \frac{2\left(\frac{66 - 6x}{7}\right) - 5}{3}$ $= \frac{\frac{132 - 12x}{7} - 5}{3}$ $= \frac{\frac{132 - 12x - 35}{7}}{3}$ $= \frac{\frac{97 - 12x}{7}}{3}$ $= \frac{97 - 12x}{21}$

7. Find $g(f^{-1}(x))$

$f^{-1}(x) = \frac{3x + 5}{2}$ $g(f^{-1}(x)) = g\left(\frac{3x + 5}{2}\right)$ $= 11 - \frac{7\left(\frac{3x + 5}{2}\right)}{6}$ $= 11 - \frac{7(3x + 5)}{12}$ $= 11 - \frac{21x + 35}{12}$ $= 11 - \frac{21x}{12} - \frac{35}{12}$ $= \frac{132 - 21x - 35}{12}$ $= \frac{97 - 21x}{12}$

Do you have any questions or need further details on any of these solutions?

Here are 5 additional questions you might ask:

1. How do you verify if two functions are inverses of each other?
2. Can you explain the concept of composite functions in more detail?
3. What are some common properties of inverse functions?
4. How can you determine if a function has an inverse?
5. What are some real-world applications of inverse functions?

Tip:

When finding the inverse of a function, always ensure to swap the dependent and independent variables and then solve for the new dependent variable. This helps in reversing the effect of the original function.