Question

directions: select all the correct locations on the graph.
a function is shown on the graph.
select all of the possible points that can be included to keep the graph a function.
(image of a graph with a red function curve/line and several blue points, along with reset and submit buttons)

Explanation:

Step1: Recall the definition of a function

A function is a relation where each input (x - value) has exactly one output (y - value). So, for a graph to represent a function, it must pass the vertical line test: no vertical line can intersect the graph more than once.

Step2: Analyze each point

• For a point to be part of the function, its x - coordinate must not have been used before (or if it is used, the y - coordinate must be the same as the existing one, but in a piece - wise and linear/non - linear graph like this, we need to check the vertical line test).
• Let's assume the points are labeled (we can infer from the graph structure). The key is that for a graph to be a function, each x has at most one y. So we check the x - values of the candidate points. The points that have unique x - values (or x - values that don't conflict with the existing graph's vertical line test) are the ones that can be included. From the visual (even though the labels are a bit unclear, the general rule is vertical line test), the points that do not cause a vertical line to intersect the graph more than once are the valid ones. After analyzing the graph (the red graph and the blue points), the points that can be included are those whose x - coordinates do not have multiple y - values in the existing graph.

Answer:

(The specific points would be determined by the vertical line test. Since the graph is not fully labeled here, but in general, the points that satisfy the vertical line test (each x has one y) are the correct ones. If we assume the standard function graph rules, the points with unique x - coordinates relative to the existing graph. For example, if we consider the graph's structure, the points that do not create a vertical line intersection. But since the original problem's graph has some blue points, the correct points are those that pass the vertical line test. If we take the common case, the points that can be included are the ones where for their x - value, there is no other point (on the red graph or other blue points) with the same x - value. So the answer depends on the exact coordinates, but following the function definition, the points that satisfy the vertical line test.)

(Note: Since the graph is not fully clear in terms of coordinates, but the process is based on the vertical line test for functions. If we had specific coordinates, we would check each point's x - coordinate against the existing graph's x - coordinates. For example, if a point has an x - value that is not present in the domain of the existing function graph, or if it has the same x - value as an existing point but the same y - value, it can be included. In the context of the given problem, after applying the vertical line test, the correct points are the ones that do not violate the function's definition.)