Question

does this graph show a function? explain how you know. a. yes; there are no y-values that have more than one x-value. b. yes; the graph passes the vertical line test. c. no; the graph fails the vertical line test. d. no; there are y-values that have more than one x-value.

Explanation:

Step1: Recall Vertical Line Test

The vertical line test states that a graph represents a function if no vertical line intersects the graph at more than one point. This is because a function must assign exactly one output (y - value) for each input (x - value).

Step2: Analyze the Graph

Looking at the given graph, when we apply the vertical line test (imagine drawing vertical lines across the graph), each vertical line will intersect the graph at most once. So the graph passes the vertical line test.

Step3: Evaluate the Options

• Option A: The definition of a function is about each x - value having at most one y - value, not y - values having at most one x - value. So A is incorrect.
• Option B: Since the graph passes the vertical line test, it represents a function. This option is correct.
• Option C: The graph does not fail the vertical line test, so C is incorrect.
• Option D: The reasoning about y - values is incorrect for the definition of a function, and the graph does represent a function. So D is incorrect.

Answer:

B. Yes; the graph passes the vertical line test.