https://file.aiquickdraw.com/mathbot/file/1727469018638ih3zol4b.jpg

The problem in the image asks us to analyze graphs of two functions, \( F(x) \) (in blue) and \( G(x) \) (in red), and then to answer some questions based on combined operations involving \( F(x) \) and \( G(x) \). The questions involve evaluating new functions defined in terms of \( F(x) \) and \( G(x) \). Let's proceed step by step to solve this.

The given graphs and questions are as follows:
- Let \( P(x) = F(x) + G(x) \)
- Let \( Q(x) = F(x) \cdot G(x) \)

We need to answer the following:
1. \( P(1) \)
2. \( Q(1) \)
3. \( P(6) \)
4. \( Q(6) \)

Let’s determine the values from the graphs.

### Step-by-Step Analysis

1. **Identify \( F(x) \) and \( G(x) \) Values from Graphs**:
   - Blue graph: \( F(x) \)
   - Red graph: \( G(x) \)

#### Finding Required Values:
1. **\( P(1) = F(1) + G(1) \)**:
   - From the blue graph, find \( F(1) \).
   - From the red graph, find \( G(1) \).
   - Add these values.

2. **\( Q(1) = F(1) \times G(1) \)**:
   - Use the values of \( F(1) \) and \( G(1) \).
   - Multiply these values.

3. **\( P(6) = F(6) + G(6) \)**:
   - From the blue graph, find \( F(6) \).
   - From the red graph, find \( G(6) \).
   - Add these values.

4. **\( Q(6) = F(6) \times G(6) \)**:
   - Use the values of \( F(6) \) and \( G(6) \).
   - Multiply these values.

#### Reading Graphs:
To provide accurate answers, I'll read the graph points:
- Blue graph (top) for \( F(x) \)
- Red graph (bottom) for \( G(x) \)

Let me analyze the image and extract these values.

### Determining Graph Values:
- **\( F(1) \)** (Blue graph):
  - It looks like \( F(1) \approx -1 \).
  
- **\( G(1) \)** (Red graph):
  - It appears that \( G(1) \approx 1 \).

So:
- \( P(1) = F(1) + G(1) = -1 + 1 = 0 \)
- \( Q(1) = F(1) \times G(1) = (-1) \times 1 = -1 \)

Now for \( x = 6 \):
- **\( F(6) \)** (Blue graph):
  - It seems \( F(6) \approx -1.5 \).

- **\( G(6) \)** (Red graph):
  - It seems \( G(6) \approx 0.5 \).

So:
- \( P(6) = F(6) + G(6) = -1.5 + 0.5 = -1 \)
- \( Q(6) = F(6) \times G(6) = (-1.5) \times 0.5 = -0.75 \)

### Final Answers:
1. \( P(1) = 0 \)
2. \( Q(1) = -1 \)
3. \( P(6) = -1 \)
4. \( Q(6) = -0.75 \)

Would you like more details or have any questions about the steps? Here are some further questions that might be useful for practice:

1. What is \( P(x) \) for other values of \( x \), like \( x = 2 \) or \( x = 4 \)?
2. How would \( Q(x) \) behave if either \( F(x) \) or \( G(x) \) were zero?
3. Could you graph \( P(x) \) and \( Q(x) \) based on these two graphs?
4. What would happen to \( P(x) \) and \( Q(x) \) if \( F(x) \) and \( G(x) \) were shifted vertically?
5. How can we interpret the behavior of \( P(x) \) when \( F(x) \) and \( G(x) \) have different signs?

**Tip**: When analyzing graphs, always identify key points and their respective function values to simplify calculations involving combinations of functions.

Analyze the graphs of F(x) (in blue) and G(x) (in red) to answer questions related to P(x) = F(x) + G(x) and Q(x) = F(x) * G(x) for specific values of x.

Learn how to solve graph-based function problems involving F(x) and G(x) to find P(x) and Q(x) for specific values of x. This problem focuses on evaluating function addition and multiplication using graphical analysis.

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