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.