The problem shows two functions, $f(x)$ (blue curve) and $g(x)$ (red curve), and asks for the identification of graphs representing $f(x) + g(x)$, $f(x) - g(x)$, and $g(x) - f(x)$ from the options labeled A to F.

Step-by-step Approach:

1. $f(x) + g(x)$: This graph will be the pointwise sum of the two functions. For every $x$, the sum is given by $f(x) + g(x)$, which generally will move each point upwards, combining the effects of both functions.

2. $f(x) - g(x)$: This graph represents the difference between the two functions. For each $x$, subtract the value of $g(x)$ from $f(x)$. Expect that at intersections where $f(x) = g(x)$, this graph will cross the x-axis.

3. $g(x) - f(x)$: This is similar to the previous one but in reverse, so expect this graph to look like a vertical reflection of $f(x) - g(x)$, crossing the x-axis at the same points but with opposite signs.

Observing the options:

• Graph A: Shows a curve that appears to represent a subtraction, as it crosses the x-axis at some points and curves downward.

• Graph B: Has a distinct curve, which could indicate a combination of the two functions, possibly representing an addition.

• Graph C, D, E, F: These show different behaviors (some crossing the x-axis, others having peaks), and we will need to match them logically to the respective operations.

Conclusion:

To determine the exact match, I will calculate the appearance of these sums and differences more carefully. Let me analyze the curves from the uploaded image to match each graph with the correct operation.I have opened the image for further analysis. Based on the visual details, I will now associate the graphs with the given functions.

Key Observations from the Graphs:

1. $f(x) + g(x)$ should combine the two curves. Since $f(x)$ and $g(x)$ have opposite curvatures (concave up and concave down), their sum should result in a flatter curve, combining their behaviors. This likely corresponds to Graph B, as it shows a combined effect without crossing the x-axis.

2. $f(x) - g(x)$ involves subtracting $g(x)$ from $f(x)$. Where the two functions intersect, the difference is zero, so this graph should cross the x-axis at those points. Based on the shapes, Graph A likely represents this function, as it crosses the axis and has the appropriate curvature.

3. $g(x) - f(x)$ is the reverse of $f(x) - g(x)$, so it should be a reflected version of the same graph. Graph C matches this, as it crosses the x-axis at the same points as Graph A but in the opposite direction.

Final Answers:

• $f(x) + g(x)$: B
• $f(x) - g(x)$: A
• $g(x) - f(x)$: C

Would you like any further details or explanations?

Follow-up Questions:

1. Why does $f(x) + g(x)$ not cross the x-axis, unlike the differences?
2. How does symmetry between $f(x) - g(x)$ and $g(x) - f(x)$ manifest visually?
3. What does the point of intersection between $f(x)$ and $g(x)$ imply for their sum and difference?
4. How would the graphs change if $g(x)$ were shifted vertically by a constant?
5. Can we predict the shape of $f(x) \times g(x)$ or $f(x) / g(x)$?

Tip:

When adding or subtracting functions, remember that the x-axis crossings of the result occur when the functions being combined are equal in magnitude but opposite in sign.