The diagram shows the graph of the sum of the functions $f(x)$ and $g(x)$ - HSC - SSCE Mathematics Extension 1 - Question 4 - 2022 - Paper 1

To solve this question, we need to analyze the graph of the sum $y = f(x) + g(x)$, which is shown in the diagram. This graph appears to be a typical upward-monotonically increasing parabola with a vertex distinctly below the x-axis. This indicates that both functions $f(x)$ and $g(x)$ are likely to have certain characteristics.

1. Identifying the Vertex: The vertex of a parabola is the point where the function reaches its maximum or minimum. Since the graph appears to open upward, it suggests that the minimum sum occurs at this vertex.

2. Analyzing the Options: Each option shows different forms of graphs. We need to check to see which pair of functions could result in the given summed function.

   • Option A: Appears incorrect as it doesn't visually align with the provided sum graph, showing intersections contradicting our graph assumptions.
   • Option B: Also incorrect since it features two functions that do not converge appropriately.
   • Option C: Shows two downward-facing curves, inappropriate for our case as they cannot yield an upward parabola through addition.
   • Option D: Displays the best representation of two functions $f(x)$ and $g(x)$ that can sum to the upward parabola as shown.

Based on our analysis, the best answer that represents the graphs of both $f(x)$ and $g(x)$ is D.