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 used 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 an oval shape centered around the origin on a coordinate grid.
3. Step 3: Apply the vertical line test. Imagine drawing vertical lines (parallel to the y-axis) across the graph. Check if any vertical line intersects the graph at more than one point.
4. Step 4: Analyze the intersections. For the oval shape, any vertical line drawn through the graph will intersect it at two points (one on the upper part of the oval and one on the lower part).
5. Step 5: Conclude based on the test. Since vertical lines intersect the graph at more than one point, the graph does not represent a function of x.

Key Concepts

Vertical Line Test

The vertical line test is a method used to determine if a graph represents a function. 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, for every x in the domain, there is a unique y in the range. Understanding this definition is crucial for identifying functions from their graphs.

Graph Interpretation

Interpreting graphs involves analyzing the visual representation of data or functions. It requires understanding the axes, scale, and shape of the graph. In the context of the vertical line test, recognizing how the graph behaves as x varies helps determine if it meets the criteria of a function.