Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 1 - Functions

Problem 1.1.23a

Chapter 1, Problem 1.1.23a

Functions and Graphs

Graph the following equations and explain why they are not graphs of functions of x.

a. |y| = x

Verified step by step guidance

First, understand the equation |y| = x. This equation involves the absolute value of y, which means y can be either positive or negative for each value of x.

To graph this equation, consider two separate cases: y = x and y = -x. These represent the two possible values of y for each x.

Graph the line y = x. This is a straight line passing through the origin with a slope of 1, meaning it rises one unit for every unit it moves to the right.

Graph the line y = -x. This is another straight line passing through the origin, but with a slope of -1, meaning it falls one unit for every unit it moves to the right.

Notice that for each x-value, there are two corresponding y-values (one positive and one negative). This violates the definition of a function, which requires each x-value to have exactly one y-value. Therefore, the graph is not a function of x.

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Key Concepts

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

Definition of a Function

A function is a relation that assigns exactly one output value for each input value. In mathematical terms, for a relation to be a function, it must pass the vertical line test, meaning that no vertical line intersects the graph of the relation at more than one point. This ensures that each input corresponds to a unique output.

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 crucial for identifying whether a given relation meets the criteria of a function.

Absolute Value Functions

An absolute value function, such as |y| = x, describes a relationship where the output is the non-negative value of the input. This specific equation implies that for each positive x, there are two corresponding y values (y and -y), which violates the definition of a function. Thus, it fails the vertical line test, indicating it is not a function of x.

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Related Practice

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Piecewise-Defined Functions

Find a formula for each function graphed in Exercises 29–32.

a. <IMAGE>

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Textbook Question

Composition of Functions

Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.

a. y = 2x − 3

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Textbook Question

In Exercises 19–32, find the (a) domain and (b) range.

________

𝔂 = 5 - √ x² - 2x - 3

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Textbook Question

Functions and Graphs

Graph the following equations and explain why they are not graphs of functions of x.