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. Step 1: Understand the vertical line test. The vertical line test is a method to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, then the graph does not represent a function.
2. Step 2: Observe the graph provided. The graph is a red curve plotted on a Cartesian plane. It appears to be continuous and smooth, without any breaks.
3. Step 3: Imagine or draw vertical lines across the graph. These lines should be parallel to the y-axis and pass through various x-values.
4. Step 4: Check the intersections of the vertical lines with the graph. For each vertical line, observe whether it intersects the graph at more than one point. If it intersects at only one point for all x-values, then the graph represents a function.
5. Step 5: Conclude based on the vertical line test. If no vertical line intersects the graph at more than one point, then y is a function of x. Otherwise, it is not a function.

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, 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 defined as a rule that assigns to each element x in X exactly one element y in Y. Understanding this definition is crucial for identifying functions from their graphs.

Graph Interpretation

Graph interpretation involves analyzing the visual representation of mathematical relationships. In the context of functions, it requires understanding how the shape and behavior of a graph reflect the relationship between x and y values. Recognizing features such as intersections, slopes, and continuity helps in determining whether a graph represents a function.