https://file.aiquickdraw.com/mathbot/file/1727468321184vagjl7n2.jpg

The problem appears to involve graphs of two functions \( F(x) \) (in blue) and \( G(x) \) (in red). You're asked to evaluate values related to the functions \( P(x) = F(x) + G(x) \) and \( Q(x) = F(x) - G(x) \) at certain points.

Specifically, you are required to find the following values:

1. \( P(1) \)  
2. \( Q(1) \)  
3. \( P(6) \)  
4. \( Q(6) \)

Let's walk through the process:

1. **Determine \( F(1) \) and \( G(1) \):**
   - Using the blue graph for \( F(x) \) and the red graph for \( G(x) \), find the values of each function at \( x = 1 \).

2. **Determine \( F(6) \) and \( G(6) \):**
   - Similarly, find the values of each function at \( x = 6 \).

3. **Compute \( P(x) \) and \( Q(x) \):**
   - \( P(x) = F(x) + G(x) \)
   - \( Q(x) = F(x) - G(x) \)

Let me analyze the graphs to extract these values for you.I have opened the image to analyze it. Now, let's extract the relevant values from the graphs.

### From the graphs:
1. **At \( x = 1 \):**
   - For \( F(1) \) (blue graph), find the y-value at \( x = 1 \).
   - For \( G(1) \) (red graph), find the y-value at \( x = 1 \).

2. **At \( x = 6 \):**
   - For \( F(6) \) (blue graph), find the y-value at \( x = 6 \).
   - For \( G(6) \) (red graph), find the y-value at \( x = 6 \).

I'll now read these values from the graph to complete the calculations for \( P(1), Q(1), P(6), \) and \( Q(6) \).Here are the values directly extracted from the graphs:

### At \( x = 1 \):
- The blue graph \( F(x) \) appears to have a y-value of **0** at \( x = 1 \).
- The red graph \( G(x) \) appears to have a y-value of **1** at \( x = 1 \).

Using these values:
- \( P(1) = F(1) + G(1) = 0 + 1 = 1 \)
- \( Q(1) = F(1) - G(1) = 0 - 1 = -1 \)

### At \( x = 6 \):
- The blue graph \( F(x) \) has a y-value of approximately **2** at \( x = 6 \).
- The red graph \( G(x) \) has a y-value of **0** at \( x = 6 \).

Using these values:
- \( P(6) = F(6) + G(6) = 2 + 0 = 2 \)
- \( Q(6) = F(6) - G(6) = 2 - 0 = 2 \)

### Final Answers:
1. \( P(1) = 1 \)
2. \( Q(1) = -1 \)
3. \( P(6) = 2 \)
4. \( Q(6) = 2 \)

Would you like more details, or do you have any questions?

### Related Questions:
1. What is the slope of the blue graph \( F(x) \) around \( x = 1 \)?
2. Can you find the value of \( P(x) \) for another \( x \)-value from the graphs?
3. How does \( G(x) \) behave as \( x \) increases beyond 6?
4. Can the behavior of \( F(x) \) be classified as exponential, linear, or another type?
5. What would \( P(x) \) and \( Q(x) \) be if \( F(x) \) and \( G(x) \) were switched?

### Tip:
When reading graphs, zooming in on key points can help ensure accuracy in determining exact values, especially for non-integer coordinates.

The graphs of the functions F(x) (in blue) and G(x) (in red) are given. You are asked to find the values of P(x) = F(x) + G(x) and Q(x) = F(x) - G(x) at x = 1 and x = 6.

Learn how to evaluate the values of P(x) = F(x) + G(x) and Q(x) = F(x) - G(x) using graphs at x = 1 and x = 6. This problem demonstrates basic operations on functions based on graphical data, suitable for high school students.

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