Let's break down the math problems based on the given image.

We are provided with the following functions:

• $f(x) = 3x + 2$
• $g(x) = 5x - 4$
• $h(x) = x^2$

The problems to solve:

1. $f(x) + g(x)$
2. $f(x) - g(x)$
3. $f(x) \cdot g(x)$
4. $(f \circ g)(x)$ (This means $f(g(x))$)
5. $(g \circ f)(x)$ (This means $g(f(x))$)
6. $(g \circ h)(x)$
7. $(f \circ g^{-1})(x)$ (This involves the inverse of $g(x)$)
8. $(g \circ f)(2)$ (Evaluate $g(f(2))$)
9. $f^{-1}(x)$ (The inverse of $f(x)$)
10. $(g^{-1} \circ h)(x)$ (This involves $g^{-1}(x)$ and applying $h(x)$)

I'll now solve these step-by-step:

Solutions:

1. $f(x) + g(x) = (3x + 2) + (5x - 4) = 8x - 2$

2. $f(x) - g(x) = (3x + 2) - (5x - 4) = 3x + 2 - 5x + 4 = -2x + 6$

3. $f(x) \cdot g(x) = (3x + 2)(5x - 4) = 15x^2 - 12x + 10x - 8 = 15x^2 - 2x - 8$

4. $(f \circ g)(x) = f(g(x)) = f(5x - 4) = 3(5x - 4) + 2 = 15x - 12 + 2 = 15x - 10$

5. $(g \circ f)(x) = g(f(x)) = g(3x + 2) = 5(3x + 2) - 4 = 15x + 10 - 4 = 15x + 6$

6. $(g \circ h)(x) = g(h(x)) = g(x^2) = 5x^2 - 4$

7. $(f \circ g^{-1})(x)$: To do this, we first need $g^{-1}(x)$.

   • $y = 5x - 4$ → Solve for $x$: $y + 4 = 5x$, $x = \frac{y + 4}{5}$
   • So, $g^{-1}(x) = \frac{x + 4}{5}$.
   • Now $(f \circ g^{-1})(x) = f\left(\frac{x + 4}{5}\right) = 3\left(\frac{x + 4}{5}\right) + 2 = \frac{3(x + 4)}{5} + 2 = \frac{3x + 12}{5} + 2 = \frac{3x + 12}{5} + \frac{10}{5} = \frac{3x + 22}{5}$

8. $(g \circ f)(2)$: First, find $f(2)$, then apply $g$.

   • $f(2) = 3(2) + 2 = 6 + 2 = 8$
   • $g(8) = 5(8) - 4 = 40 - 4 = 36$

9. $f^{-1}(x)$: To find the inverse of $f(x) = 3x + 2$, solve for $x$.

   • $y = 3x + 2$
   • $y - 2 = 3x$
   • $x = \frac{y - 2}{3}$
   • So, $f^{-1}(x) = \frac{x - 2}{3}$

10. $(g^{-1} \circ h)(x)$: First apply $h(x)$, then $g^{-1}$.

• $h(x) = x^2$
• Now, apply $g^{-1}(x^2) = \frac{x^2 + 4}{5}$

Let me know if you need more details or have any questions.

Here are 5 additional questions to extend your understanding:

1. What is the difference between the composition of functions and the sum of functions?
2. How do you derive the inverse of a linear function like $f(x) = 3x + 2$?
3. How does composition affect the domain and range of functions like $f(x)$ and $g(x)$?
4. What are the implications of applying the inverse of a function before another function in composition?
5. How can you check if two functions are inverses of each other?

Tip: Always remember to apply functions in the correct order when performing compositions, especially when inverses are involved.