https://file.aiquickdraw.com/mathbot/file/1725928332802y9ihsfkw.jpg

To solve the functions problem from the image, I will break down each question one by one based on the given functions and the formulas provided:

Given functions:
- \( f(x) = 2x - 1 \)
- \( r(x) = x + 3 \)
- \( g(x) = |3x - 4| \)
- \( h(x) = \frac{x}{2} \)
- \( s(x) = x^2 - 4x - 21 \)

### Column A:
1. **\( f(0) \)**:
   - \( f(x) = 2x - 1 \)
   - \( f(0) = 2(0) - 1 = -1 \)

2. **\( g(3) \)**:
   - \( g(x) = |3x - 4| \)
   - \( g(3) = |3(3) - 4| = |9 - 4| = |5| = 5 \)

3. **\( s(-1) \)**:
   - \( s(x) = x^2 - 4x - 21 \)
   - \( s(-1) = (-1)^2 - 4(-1) - 21 = 1 + 4 - 21 = -16 \)

4. **\( h(0) \)**:
   - \( h(x) = \frac{x}{2} \)
   - \( h(0) = \frac{0}{2} = 0 \)

5. **\( (f + r)(x) \)**:
   - \( f(x) + r(x) = (2x - 1) + (x + 3) = 3x + 2 \)

6. **\( (f + r)(3) \)**:
   - \( (f + r)(x) = 3x + 2 \)
   - \( (f + r)(3) = 3(3) + 2 = 9 + 2 = 11 \)

7. **\( (r - f)(x) \)**:
   - \( r(x) - f(x) = (x + 3) - (2x - 1) = x + 3 - 2x + 1 = -x + 4 \)

8. **\( (r - f)(2) \)**:
   - \( (r - f)(x) = -x + 4 \)
   - \( (r - f)(2) = -(2) + 4 = -2 + 4 = 2 \)

9. **\( (f \cdot r)(x) \)**:
   - \( f(x) \cdot r(x) = (2x - 1)(x + 3) \)
   - Expand the product:
     \[
     (2x - 1)(x + 3) = 2x^2 + 6x - x - 3 = 2x^2 + 5x - 3
     \]

10. **\( (f \cdot r)(1) \)**:
    - \( (f \cdot r)(x) = 2x^2 + 5x - 3 \)
    - \( (f \cdot r)(1) = 2(1)^2 + 5(1) - 3 = 2 + 5 - 3 = 4 \)

11. **\( \frac{s}{r}(x) \)**:
    - \( \frac{s(x)}{r(x)} = \frac{x^2 - 4x - 21}{x + 3} \)
    - Perform polynomial division or factorization:
      \[
      x^2 - 4x - 21 = (x + 3)(x - 7)
      \]
      So,
      \[
      \frac{s(x)}{r(x)} = x - 7
      \]

### Column B (Answers from matching computations):
- A. \(-11\)
- B. 2
- C. \( 3x + 2 \)
- D. \( x - 7 \)
- E. \( -x + 4 \)
- F. 0
- G. \( 2x^2 + 5x - 3 \)
- H. 6
- I. \(-16\)
- J. \( 2x + 2 \)
- K. 5

### Matching:
1. \( f(0) = -1 \) (Not in the options, but we'll proceed)
2. \( g(3) = 5 \) → Match with **K**
3. \( s(-1) = -16 \) → Match with **I**
4. \( h(0) = 0 \) → Match with **F**
5. \( (f + r)(x) = 3x + 2 \) → Match with **C**
6. \( (f + r)(3) = 11 \) → (Not listed)
7. \( (r - f)(x) = -x + 4 \) → Match with **E**
8. \( (r - f)(2) = 2 \) → Match with **B**
9. \( (f \cdot r)(x) = 2x^2 + 5x - 3 \) → Match with **G**
10. \( (f \cdot r)(1) = 4 \) (Not listed, but could be **H**)
11. \( \frac{s}{r}(x) = x - 7 \) → Match with **D**

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Let me know if you would like any further details or clarification!

Here are five related questions for further practice:
1. What is the derivative of \( f(x) = 2x - 1 \)?
2. How do you solve polynomial division in general?
3. What is the graphical interpretation of \( g(x) = |3x - 4| \)?
4. How do you find the roots of \( s(x) = x^2 - 4x - 21 \)?
5. What is the significance of absolute value functions?

**Tip:** Always check if polynomial expressions can be factored to simplify division or other operations easily.

Answer operations on functions problems using given formulas: f(x) = 2x - 1, g(x) = |3x - 4|, h(x) = x/2, r(x) = x + 3, s(x) = x^2 - 4x - 21.

Explore detailed solutions to algebraic operations on functions involving polynomials, absolute values, and simple linear expressions. Suitable for high school students learning advanced algebra concepts.

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