Question

which statement is a correct interpretation of the vertical line test?

if only one vertical line intersects the graph at exactly one point, the graph represents a function.
if only one vertical line intersects the graph at exactly one point, the graph does not represent a function.
if any vertical line can intersect the graph at more than one point, the graph represents a function.
if any vertical line can intersect the graph at more than one point, the graph does not represent a function.

Explanation:

Define a mathematical function

A relation is a function if each input \(x\) maps to exactly one output \(y\).

Understand the vertical line test

The vertical line test is a visual way to determine if a curve on a graph represents a function. A vertical line represents a single input value \(x\). If any vertical line intersects the graph at more than one point, it means that single input \(x\) has multiple outputs \(y\), violating the definition of a function.

Analyze the given statements

• Statement 1: "If only one vertical line intersects the graph at exactly one point, the graph represents a function." This is incorrect because every possible vertical line must intersect the graph at most once, not just "only one" line.
• Statement 2: "If only one vertical line intersects the graph at exactly one point, the graph does not represent a function." This is incorrect.
• Statement 3: "If any vertical line can intersect the graph at more than one point, the graph represents a function." This is incorrect because intersecting at more than one point means it is not a function.
• Statement 4: "If any vertical line can intersect the graph at more than one point, the graph does not represent a function." This is correct because it directly matches the definition of failing the vertical line test.

Answer:

• (A) If only one vertical line intersects the graph at exactly one point, the graph represents a function.
• (B) If only one vertical line intersects the graph at exactly one point, the graph does not represent a function.
• (C) If any vertical line can intersect the graph at more than one point, the graph represents a function.
• (D) If any vertical line can intersect the graph at more than one point, the graph does not represent a function. (Correct answer)