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: Analyze the blue curve. Observe the blue curve on the graph. Draw imaginary vertical lines across the graph. Notice that each vertical line intersects the blue curve at most one point. This indicates that the blue curve represents a function.
3. Step 3: Analyze the orange curve. Observe the orange curve on the graph. Draw imaginary vertical lines across the graph. Notice that each vertical line intersects the orange curve at most one point. This indicates that the orange curve also represents a function.
4. Step 4: Confirm the results. Since both the blue and orange curves pass the vertical line test, we can conclude that y is a function of x for both curves.
5. Step 5: Summarize the findings. Both the blue and orange curves represent functions because they pass the vertical line test, meaning no vertical line intersects either curve at more than one point.

Key Concepts

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-value in the domain, there is a unique y-value in the range. Understanding this definition is crucial for determining whether a graph represents a function.

Vertical Line Test

The vertical line test is a method used to determine if a graph represents a function. If any vertical line drawn through the graph intersects it at more than one point, the graph does not represent a function. This test is essential for visually assessing the relationship between x and y values in a graph.

Graph Interpretation

Interpreting graphs involves analyzing the visual representation of data to understand the relationships between variables. In the context of functions, it requires recognizing how changes in x affect y, and identifying key features such as intercepts, asymptotes, and the overall shape of the graph. This skill is vital for applying the vertical line test effectively.