Book cover for Precalculus: A Right Triangle Approach

Precalculus: A Right Triangle Approach

Judith A. Beecher, Marvin L. Bittinger, David J. Ellenbogen, Judith A. Penna

ISBN #9780321783967

5th Edition

5,905 Questions

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25,202 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section delves into the behaviors of functions by defining and determining increasing, decreasing, and constant intervals, as well as locating relative maximums and minimums on function graphs. It introduces piecewise functions, including the greatest integer function, emphasizing careful attention to domain restrictions and use of open interval notation. Real-world applications such as car distance problems and area optimization illustrate how these mathematical concepts are applied in practice.

Learning Objectives

1

Identify and determine the intervals on which a function is increasing, decreasing, or constant.

2

Analyze function graphs to locate relative maximums and minimums using open interval notation.

3

Construct and evaluate piecewise functions, including the greatest integer function.

4

Apply function concepts to real-world problems such as car distance and area optimization.

Key Concepts

CONCEPT

DEFINITION

Increasing Function

A function f is increasing on an open interval I if for any a and b in I with a < b, f(a) < f(b).

Decreasing Function

A function f is decreasing on an open interval I if for any a and b in I with a < b, f(a) > f(b).

Constant Function

A function is constant on an open interval I if for any a and b in I, f(a) = f(b).

Relative Maximum

A function has a relative maximum at c if there exists an open interval containing c such that for all x in that interval, f(c) ≥ f(x).

Relative Minimum

A function has a relative minimum at c if there exists an open interval containing c such that for all x in that interval, f(c) ≤ f(x).

Piecewise Function

A function defined by different expressions over different parts of its domain.

Greatest Integer Function

Denoted by ⌊x⌋, it assigns to each real number x the greatest integer less than or equal to x and is defined piecewise.

Example Problems

Example 1

Determine the intervals on which the function is (a) increasing, (b) decreasing, and (c) constant.

Example 2

Determine the intervals on which the function is (a) increasing, (b) decreasing, and (c) constant.

Example 3

Determine the intervals on which the function is (a) increasing, (b) decreasing, and (c) constant.

Example 4

Determine the intervals on which the function is (a) increasing, (b) decreasing, and (c) constant.

Example 5

Determine the intervals on which the function is (a) increasing, (b) decreasing, and (c) constant.
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Step-by-Step Explanations

QUESTION

Given a function graph, how do you determine the interval where the function is increasing?

STEP-BY-STEP ANSWER:

Step 1: Identify the domain of the function; only consider values where the function is defined.
Step 2: Examine the graph to see where, as x increases, the corresponding y-values also increase.
Step 3: Mark the open interval (a, b) over which the y-values show a consistent upward trend.
Final Answer: The function is increasing on the interval (a, b) if for every a < x < b, f(a) < f(x).

Determining Increasing Intervals

QUESTION

How do you determine if a function has a relative maximum or minimum at a point?

STEP-BY-STEP ANSWER:

Step 1: Identify critical points on the graph where the function changes direction (peaks and valleys).
Step 2: For a relative maximum, check that the function value at the point is higher than all other nearby values; for a minimum, it must be lower.
Step 3: Confirm using open intervals around the point to avoid including endpoints where slope might be ambiguous.
Final Answer: A point c is a relative maximum if there exists an open interval I containing c such that f(c) ≥ f(x) for all x in I. The reverse holds for a relative minimum.

Finding Relative Maxima and Minima

QUESTION

How do you evaluate a piecewise function at a specific x-value?

STEP-BY-STEP ANSWER:

Step 1: Determine which condition of the piecewise definition the x-value satisfies.
Step 2: Use the corresponding formula for that piece of the function.
Step 3: Substitute the x-value into the selected expression.
Final Answer: The function’s output is given by the formula corresponding to the interval in which the x-value lies.

Evaluating Piecewise Functions

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Common Mistakes

  • Assuming a function can be both increasing and decreasing at a single point, rather than using open intervals to clearly define behavior.
  • Incorrectly including endpoints when determining increasing or decreasing intervals, leading to conflicting interpretations.
  • Neglecting domain restrictions when evaluating piecewise functions, which can lead to errors in function evaluation.
  • Overlooking the need to use the positive square root when applying the Pythagorean theorem in real-world applications.