Ch. 1 - Functions

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

Hass 15th Edition

Ch. 1 - Functions

Problem 1.1.8b

Chapter 1, Problem 1.1.8b

Functions

In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.

b. <IMAGE>

Verified step by step guidance

Step 1: Understand the definition of a function. A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

Step 2: Use the Vertical Line Test to determine if a graph represents a function. The Vertical Line Test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.

Step 3: Examine the given graph (represented by ) and imagine drawing vertical lines across the entire graph.

Step 4: Check if any vertical line intersects the graph at more than one point. If it does, the graph is not a function of x. If every vertical line intersects the graph at most once, then it is a function of x.

Step 5: Provide a reasoned conclusion based on the Vertical Line Test. If the graph passes the test, state that it is a function of x. If it fails, explain that it is not a function of x because there exists at least one vertical line that intersects the graph at more than one point.

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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 between a set of inputs and a set of possible outputs where each input is related to exactly one output. This means that 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 given 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 provides a visual way to assess the uniqueness of outputs for each input.

Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). Understanding the domain and range helps in analyzing the behavior of functions and is essential for identifying whether a graph meets the criteria of a function.

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Finding the Domain and Range of a Graph

Related Practice

Textbook Question

The accompanying figure shows the graph of a function f(x) with domain [0,2] and range [0,1]. Find the domains and ranges of the following functions, and sketch their graphs.

<IMAGE>

c. 2f(x)

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Textbook Question

The Greatest and Least Integer Functions

For what values of x is

b. ⌈x⌉ = 0

Textbook Question

Industrial costs A power plant sits next to a river where the river is 800 ft wide. Laying a new cable from the plant to a location in the city 2 mi downstream on the opposite side costs \$180 per foot across the river and \$100 per foot along the land.

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b. Generate a table of values to determine whether the least expensive location for point Q is less than 2000 ft or greater than 2000 ft from point P.

Textbook Question

Composition of Functions

Evaluate each expression using the functions

f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0

x − 1, 0 ≤ x ≤ 2

c. g(g(−1))

Textbook Question

Find the largest interval on which the given function is increasing.

b. ƒ(x) = (x + 1)⁴

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Textbook Question

Piecewise-Defined Functions