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The problem asks you to analyze the graphs of two functions \( f(x) \) and \( g(x) \), sketched in blue and red, respectively. Based on the graphs, you are asked to match the resulting expressions for combinations of \( f(x) \) and \( g(x) \) with corresponding graphs (labeled A-F).

The functions provided are:
1. \( f(x) + g(x) \)
2. \( f(x) - g(x) \)
3. \( g(x) - f(x) \)

Let's break this down step by step:

1. **Analyzing \( f(x) + g(x) \):**
   - When adding \( f(x) \) and \( g(x) \), you're combining the values of the two functions at each point. This means the resulting graph will be higher than either function at points where both \( f(x) \) and \( g(x) \) are positive, and lower when both are negative.
   - Looking at graph **B**, it shows a curve that is higher than either individual function around the center, which suggests it could represent \( f(x) + g(x) \).

2. **Analyzing \( f(x) - g(x) \):**
   - For this operation, you're subtracting \( g(x) \) from \( f(x) \). The result will depend on the relative size of \( f(x) \) and \( g(x) \). Where \( f(x) \) is greater than \( g(x) \), the graph will be positive, and where \( g(x) \) is greater, it will be negative.
   - Based on this, graph **A** seems to show this behavior as it shows a curve that remains mostly positive where \( f(x) \) is greater.

3. **Analyzing \( g(x) - f(x) \):**
   - This is simply the reverse of the previous operation. Instead of subtracting \( g(x) \) from \( f(x) \), you are subtracting \( f(x) \) from \( g(x) \). The shape should be a mirror image of the previous result.
   - Graph **E** fits this description, showing a curve that is inverted compared to graph **A**, making it a match for \( g(x) - f(x) \).

Thus, the correct pairings are:
- \( f(x) + g(x) \) matches graph **B**.
- \( f(x) - g(x) \) matches graph **A**.
- \( g(x) - f(x) \) matches graph **E**.

Would you like more details on any of these operations or the reasoning process?

Here are 5 related questions for further exploration:
1. What happens to the graph when you multiply \( f(x) \) and \( g(x) \)?
2. How would the graph of \( f(x) \times g(x) \) compare to their sum or difference?
3. How would shifting \( f(x) \) or \( g(x) \) horizontally affect their combined graph?
4. What would the graph look like if you divide \( f(x) \) by \( g(x) \)?
5. Can you deduce the x-intercepts of the sum or difference of the functions based on their individual x-intercepts?

**Tip:** Always focus on how the values of the individual functions behave at key points (such as zeros, maxima, and minima) when performing operations like addition and subtraction.

Consider the graphs of y = f(x) and y = g(x) sketched in blue and red respectively. Match the expressions f(x) + g(x), f(x) - g(x), and g(x) - f(x) with the corresponding graphs A-F.

Analyze the graphs of two functions, f(x) and g(x), and match their sum and difference with the correct graphs. This problem focuses on understanding how the addition and subtraction of functions affect their graphical representations. Suitable for Grades 10-12.

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