https://file.aiquickdraw.com/mathbot/file/1728613172351jr2w45av.jpg

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 \( g^{-1}(x^2) = \frac{x^2 + 4}{5} \)

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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.

Mid Semester Exam (Mathematics, Grade XI, 2024/2025): Solve the following based on the functions f(x) = 3x + 2, g(x) = 5x - 4, h(x) = x^2. Questions include addition, subtraction, multiplication of functions, composition of functions, and finding inverses.

Solve mathematical problems based on the functions f(x) = 3x + 2, g(x) = 5x - 4, and h(x) = x^2. Includes function addition, subtraction, composition, and finding inverses. Suitable for Grade 11 students, covering key algebra concepts and operations.

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