Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 12

Chapter 3, Problem 12

Use the vertical line test to identify graphs in which y is a function of x.

Verified step by step guidance

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.

Observe the graph provided. It is a circle centered at the origin with a radius of approximately 3 units.

Imagine drawing vertical lines (parallel to the y-axis) across the graph. For example, consider a vertical line at x = 1. This line intersects the circle at two points: one above the x-axis and one below the x-axis.

Since there exists at least one vertical line that intersects the graph at more than one point, the graph does not pass the vertical line test.

Conclude that the graph does not represent y as a function of x because it fails the vertical line test.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 of x. 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. In the context of the vertical line test, it requires recognizing how the shape of the graph (like a circle) affects the relationship between x and y. This skill is essential for determining whether a graph meets the criteria of a function.

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In Exercises 11–26, determine whether each equation defines y as a function of x. x + y = 16

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Use the graph of y = f(x) to graph each function g.

g(x) = f(x/2)

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Use the graph of y = f(x) to graph each function g.

g(x) = 2f(x)

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Find the domain of each function. f(x) = 1/(x²+1) - 1/(x²-1)

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