Book cover for Thomas Calculus

Thomas Calculus

George B. Thomas, Jr.

ISBN #9780321878960

13th Edition

6,812 Questions

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127,035 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes the central role of functions in mathematics, illustrating how they model real-world relationships. Key concepts include understanding domain and range, graphing functions, applying the vertical line test, and exploring various types of functions such as polynomial, trigonometric, and piecewise-defined functions. Additionally, the combination and transformation of functions enable us to develop more complex models in calculus.

Learning Objectives

1

Describe what a function is and explain how to determine its domain and range.

2

Interpret and sketch graphs of functions and apply the vertical line test to verify functional relationships.

3

Differentiate among various types of functions such as polynomial, power, piecewise-defined, trigonometric, exponential, and logarithmic functions.

4

Develop and analyze function combinations including sums, differences, products, and quotients, as well as function transformations.

5

Recognize common errors and misconceptions when working with functions and their graphical representations.

Key Concepts

CONCEPT

DEFINITION

Function

A rule that assigns each element x in a set D (domain) a unique element f(x) in a set Y (range).

Domain

The set of all possible input values for which the function is defined.

Range

The set of all output values produced by the function as x varies over the domain.

Vertical Line Test

A method to determine if a curve in the coordinate plane represents a function; a vertical line must intersect the graph at most once.

Piecewise-Defined Function

A function that is defined by different formulas over different parts of its domain.

Even Function

A function f(x) is even if f(-x) = f(x) for all x in the domain; its graph is symmetric about the y-axis.

Odd Function

A function f(x) is odd if f(-x) = -f(x) for all x in the domain; its graph is symmetric about the origin.

Function Combination

Operations such as addition, subtraction, multiplication, and division performed on functions to create new functions.

Example Problems

Example 1

In Exercises $1-6,$ find the domain and range of each function. $$f(x)=1+x^{2}$$

Example 2

In Exercises $1-6,$ find the domain and range of each function. $$f(x)=1-\sqrt{x}$$

Example 3

In Exercises $1-6,$ find the domain and range of each function. $$ F(x)=\sqrt{5 x+10} $$

Example 4

In Exercises $1-6,$ find the domain and range of each function. $$g(x)=\sqrt{x^{2}-3 x}$$

Example 5

In Exercises $1-6,$ find the domain and range of each function. $$f(t)=\frac{4}{3-t}$$
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Step-by-Step Explanations

QUESTION

For the function f(x) = x², what are the natural domain and range?

STEP-BY-STEP ANSWER:

Step 1: Identify the formula f(x) = x². Since squaring any real number is defined, the domain is all real numbers, (-∞, ∞).
Step 2: Determine the range by noticing that x² is always nonnegative. Thus, the range is [0, ∞).
Final Answer: Domain = (-∞, ∞) and Range = [0, ∞).

Determining the Domain and Range of a Function

QUESTION

How can you verify if the graph of a circle represents the graph of a function?

STEP-BY-STEP ANSWER:

Step 1: Understand that a function must assign exactly one output for each input x.
Step 2: Draw vertical lines across the graph of the circle.
Step 3: Observe that many vertical lines intersect the circle at two points.
Final Answer: Since some vertical lines intersect the circle in more than one point, the circle is not the graph of a function.

Applying the Vertical Line Test

QUESTION

Given f(x) = 2x and g(x) = 21 - x, how do you form and determine the domain of the function (f + g)(x)?

STEP-BY-STEP ANSWER:

Step 1: Write the formula: (f + g)(x) = f(x) + g(x) = 2x + (21 - x) = x + 21.
Step 2: Identify the domain; since both f and g are defined on their respective domains and the common domain is where both are defined, assume both are defined for x in (0, ∞) if stated, or typically for all real numbers if un-restricted.
Step 3: In this textbook example, the domains were given as intersecting intervals. Assuming f has domain (0, ∞) and g has domain (-∞, 14), the common domain would be (0, 14).
Final Answer: (f + g)(x) = x + 21 with domain = (0, 14) (based on the given context).

Combining Functions

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Common Mistakes

  • Assuming that any curve drawn is the graph of a function without applying the vertical line test.
  • Overlooking restrictions on the domain when a function’s formula implies limitations (e.g., square roots or division by zero).
  • Confusing the properties of even and odd functions, especially when small adjustments (like adding a constant) alter symmetry.
  • Treating piecewise-defined functions as separate functions rather than one unified function defined over different intervals.