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### Question 8:
Given:
- \( f(x) = \frac{x+3}{2x-7} \)
- \( g(x) = \frac{2x-3}{x+1} \)
- \( h(x) = \frac{5x+1}{3x-1} \)

You are asked to:
1. **Evaluate**:
   - \( f(5) \)
   - \( g(4) \)
2. **Show** that:
   - \( h(f(5)) = \frac{43}{21} \)

### Solution:
#### (a) Evaluate \( f(5) \) and \( g(4) \):
1. **For \( f(5) \)**:
   \[
   f(x) = \frac{x+3}{2x-7}
   \]
   Substitute \( x = 5 \):
   \[
   f(5) = \frac{5 + 3}{2(5) - 7} = \frac{8}{10 - 7} = \frac{8}{3}
   \]

2. **For \( g(4) \)**:
   \[
   g(x) = \frac{2x - 3}{x + 1}
   \]
   Substitute \( x = 4 \):
   \[
   g(4) = \frac{2(4) - 3}{4 + 1} = \frac{8 - 3}{5} = \frac{5}{5} = 1
   \]

#### (b) Show that \( h(f(5)) = \frac{43}{21} \):
- First, recall that \( f(5) = \frac{8}{3} \).
- Now, substitute \( f(5) \) into the function \( h(x) \):
  \[
  h(x) = \frac{5x+1}{3x-1}
  \]
  Substitute \( x = \frac{8}{3} \) into \( h(x) \):
  \[
  h\left(\frac{8}{3}\right) = \frac{5\left(\frac{8}{3}\right) + 1}{3\left(\frac{8}{3}\right) - 1} = \frac{\frac{40}{3} + 1}{8 - 1} = \frac{\frac{40}{3} + \frac{3}{3}}{7} = \frac{\frac{43}{3}}{7}
  \]
  Simplifying further:
  \[
  h\left(\frac{8}{3}\right) = \frac{43}{21}
  \]
Thus, it is shown that \( h(f(5)) = \frac{43}{21} \).

---

### Question 17:
Given:
- \( f(x) = 3x - 1 \)
- \( g(x) = 2x \)
- \( h(x) = x + 1 \)

You are asked to:
1. Find \( h(g(f(x))) \)
2. Find \( h(g(f(-2))) \)

### Solution:
#### (a) Find \( h(g(f(x))) \):
1. First, find \( f(x) \):
   \[
   f(x) = 3x - 1
   \]
2. Now, find \( g(f(x)) \):
   \[
   g(x) = 2x \quad \text{so} \quad g(f(x)) = 2(3x - 1) = 6x - 2
   \]
3. Finally, find \( h(g(f(x))) \):
   \[
   h(x) = x + 1 \quad \text{so} \quad h(g(f(x))) = (6x - 2) + 1 = 6x - 1
   \]

#### (b) Find \( h(g(f(-2))) \):
1. First, calculate \( f(-2) \):
   \[
   f(-2) = 3(-2) - 1 = -6 - 1 = -7
   \]
2. Now, calculate \( g(f(-2)) \):
   \[
   g(-7) = 2(-7) = -14
   \]
3. Finally, calculate \( h(g(f(-2))) \):
   \[
   h(-14) = -14 + 1 = -13
   \]

---

Would you like to dive deeper into any part of the explanation, or do you have any questions?

### Related Questions:
1. How would the result change if \( f(x) \), \( g(x) \), or \( h(x) \) were modified?
2. Can you evaluate \( h(g(f(1))) \) using the same method?
3. How would you approach finding \( g(h(f(x))) \)?
4. What happens to the compositions if the domain of one function is restricted?
5. Can you derive the inverse of \( f(x) \) or \( g(x) \)?

### Tip:
When working with composite functions, always start from the innermost function and work your way outward step by step.

Given f(x) = (x+3)/(2x-7), g(x) = (2x-3)/(x+1), h(x) = (5x+1)/(3x-1), evaluate f(5) and g(4) and show that h(f(5)) = 43/21. Also, given f(x) = 3x-1, g(x) = 2x, and h(x) = x+1, find h(g(f(x))) and h(g(f(-2))).

Learn how to evaluate complex functions and compositions such as f(x) = (x+3)/(2x-7), g(x) = (2x-3)/(x+1), and h(x) = (5x+1)/(3x-1). This includes step-by-step evaluations and verification, ideal for high school students (Grades 10-12).

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