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The problem asks to determine the values of \( A \), \( B \), \( C \), and \( D \) for the function \( g(x) = 2f(x) + 5 \), based on the provided table for \( f(x) \).

The values of \( f(x) \) at different points \( x = -2, 0, 1, 5 \) are given, and we are asked to compute the corresponding values of \( g(x) \) using the equation \( g(x) = 2f(x) + 5 \).

### Step-by-step Calculation:
1. **For \( x = -2 \):**
   - \( f(-2) = -4 \)
   - \( g(-2) = 2f(-2) + 5 = 2(-4) + 5 = -8 + 5 = -3 \)
   - Therefore, \( A = -3 \).

2. **For \( x = 0 \):**
   - \( f(0) = -1 \)
   - \( g(0) = 2f(0) + 5 = 2(-1) + 5 = -2 + 5 = 3 \)
   - Therefore, \( B = 3 \).

3. **For \( x = 1 \):**
   - \( f(1) = 3 \)
   - \( g(1) = 2f(1) + 5 = 2(3) + 5 = 6 + 5 = 11 \)
   - Therefore, \( C = 11 \).

4. **For \( x = 5 \):**
   - \( f(5) = 7 \)
   - \( g(5) = 2f(5) + 5 = 2(7) + 5 = 14 + 5 = 19 \)
   - Therefore, \( D = 19 \).

### Final Answers:
- \( A = -3 \)
- \( B = 3 \)
- \( C = 11 \)
- \( D = 19 \)

Would you like a more detailed explanation on any step? Here are some related questions for further practice:

1. If \( g(x) = 3f(x) + 4 \), what would \( A \), \( B \), \( C \), and \( D \) be?
2. What happens to \( g(x) \) if \( f(x) \) is multiplied by a different constant?
3. Can you describe how the graph of \( g(x) \) relates to the graph of \( f(x) \)?
4. How would you solve if the function \( g(x) \) was given as \( g(x) = f(x) - 2 \)?
5. If \( g(x) \) was a quadratic function, what would the new values of \( A \), \( B \), \( C \), and \( D \) be?

**Tip:** When modifying functions, always check if a transformation affects the slope, the intercept, or both.

Suppose a function f(x) is defined by the given coordinates in the table below. Determine the values A, B, C, and D in the table for the function g(x) = 2f(x) + 5.

Learn how to calculate values of A, B, C, and D for the function g(x) = 2f(x) + 5. A step-by-step solution is provided for each x-value using basic function transformations. Suitable for students in Grades 9-10.

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