In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.

Verified step by step guidance

1. Understand the vertical line test: The vertical line test is a method used to determine if a graph represents a function. A graph represents a function if and only if no vertical line intersects the graph at more than one point.
2. Examine the graph: Look at the graph provided in the exercise. Imagine or draw vertical lines (parallel to the y-axis) across the graph.
3. Apply the test: For each vertical line, check if it intersects the graph at more than one point. If any vertical line intersects the graph at more than one point, the graph does not represent a function.
4. Conclude based on the test: If no vertical line intersects the graph at more than one point, then the graph represents a function. Otherwise, it does not.
5. Summarize your findings: State whether the graph passes or fails the vertical line test and explain why, based on the intersections observed.

Key Concepts

Vertical Line Test

The vertical line test is a method used to determine if a graph represents a function. According to this test, if any vertical line intersects the graph at more than one point, then the graph does not represent a function. This is because a function must assign exactly one output (y-value) for each input (x-value).

Function Definition

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In mathematical terms, a function f from a set X to a set Y is denoted as f: X → Y, meaning for every x in X, there is a unique y in Y. Understanding this definition is crucial for identifying functions in graphical representations.

Graphical Representation of Functions

Graphs visually represent functions, showing the relationship between the independent variable (x) and the dependent variable (y). Each point on the graph corresponds to a pair (x, y). Recognizing how different types of graphs (linear, quadratic, etc.) behave helps in applying the vertical line test effectively to determine if they represent functions.