What does the Vertical Line Test determine about a relation?

What does the Vertical Line Test determine about a relation?

• A. If every vertical line intersects the graph at exactly one point.
• B. If the graph is symmetrical with respect to the x-axis.
• C. If the graph is continuous for all x.
• D. If a vertical line intersects the graph in at most one point.

Answer: (D)

A relation is a function when each input x has exactly one output y. The vertical line test checks this by seeing how many points the graph shares with a vertical line x = c. If every vertical line hits the graph at most one point, then no x-value corresponds to more than one y-value, so the relation is a function. If a vertical line ever crosses the graph more than once, that x would give two different y-values, which isn’t allowed for a function. That’s why the best description is: a vertical line intersects the graph in at most one point. The other statements aren’t about the function property: symmetry about the x-axis is about reflection, continuity for all x isn’t required for a function, and insisting on exactly one intersection is too strong since some x-values may not be in the domain.

A relation is a function when each input x has exactly one output y. The vertical line test checks this by seeing how many points the graph shares with a vertical line x = c. If every vertical line hits the graph at most one point, then no x-value corresponds to more than one y-value, so the relation is a function. If a vertical line ever crosses the graph more than once, that x would give two different y-values, which isn’t allowed for a function.

That’s why the best description is: a vertical line intersects the graph in at most one point. The other statements aren’t about the function property: symmetry about the x-axis is about reflection, continuity for all x isn’t required for a function, and insisting on exactly one intersection is too strong since some x-values may not be in the domain.