Question

which of the following graphs represents a one-to-one function?

Explanation:

Step1: Recall One - to - One Test

A function is one - to - one if no horizontal line intersects its graph more than once (Horizontal Line Test). Also, first, we can check if the graph is a function (Vertical Line Test: a graph is a function if no vertical line intersects it more than once).

Step2: Analyze the First Graph

For the first graph, we apply the Vertical Line Test: any vertical line will intersect the graph at most once, so it is a function. Now apply the Horizontal Line Test: as we move a horizontal line across the graph, it will intersect the graph at most once. The graph has a part that is increasing and some oscillating parts, but overall, when we check horizontal lines, each horizontal line intersects the graph at only one point.

Step3: Analyze the Second Graph

The second graph is a downward - opening parabola (or a "hill - shaped" graph). For a vertical line test, it is a function (since vertical lines intersect at most once). But for the horizontal line test, a horizontal line (e.g., \(y = 3.5\)) will intersect the graph at two points (left and right of the vertex), so it fails the Horizontal Line Test and is not one - to - one.

Answer:

The graph (the first one shown in the problem, the upper graph with the oscillating and increasing part) represents a one - to - one function.