BackPrecalculus: Increasing, Decreasing, and Piecewise Functions
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Section 2.1: Increasing, Decreasing, and Piecewise Functions; Applications
Objectives
To graph functions and identify intervals where the function is increasing, decreasing, or constant, and to estimate relative maxima and minima.
To model real-world applications with functions, determine their domains and values, and graph them.
To graph and analyze piecewise-defined functions.
Increasing, Decreasing, and Constant Functions
Definitions and Graphical Interpretation
Understanding how a function behaves on an interval is fundamental in precalculus. The behavior can be classified as increasing, decreasing, or constant.
Increasing Function: A function f is increasing on an open interval I if, for any two numbers a and b in I with a < b, it follows that f(a) < f(b).
Decreasing Function: A function f is decreasing on an open interval I if, for any a < b in I, f(a) > f(b).
Constant Function: A function f is constant on an open interval I if, for any a and b in I, f(a) = f(b).
These behaviors can be visualized on a graph:
Increasing: The graph rises from left to right.
Decreasing: The graph falls from left to right.
Constant: The graph is a horizontal line.
Relative Maximum and Minimum Values
Definitions
Relative (or local) extrema are important for understanding the peaks and valleys of a function.
Relative Maximum: f(c) is a relative maximum if there exists an open interval containing c such that f(c) > f(x) for all x in the interval, where x ≠ c.
Relative Minimum: f(c) is a relative minimum if there exists an open interval containing c such that f(c) < f(x) for all x in the interval, where x ≠ c.
These points are often found at the "turning points" of the graph.
Applications: Modeling with Functions
Example: Distance Between Two Moving Objects
Suppose two nurses, Kiara and Matias, drive away from a hospital at right angles to each other. Kiara’s speed is 35 mph and Matias’s is 40 mph. We are to express the distance between the cars as a function of time, d(t), and find the domain of the function.
After t hours, Kiara has traveled 35t miles and Matias 40t miles.
Since they travel at right angles, the distance between them forms the hypotenuse of a right triangle with legs 35t and 40t.
By the Pythagorean Theorem:
Domain: Since time cannot be negative, t ≥ 0. Thus, the domain is .
Piecewise-Defined Functions
Definition and Evaluation
A piecewise-defined function uses different formulas for different parts of its domain. Each "piece" applies to a specific interval.
Example function:
, for
, for
, for
To evaluate at various points:
(since )
(since )
(since )
(since ; note: if is not included, use the next piece)
(since )
(since )
Graphing Piecewise Functions
Graph each "piece" on its respective interval.
Use open or closed circles to indicate whether endpoints are included.
Discontinuities may occur at the boundaries between pieces.
Example: For for , and for , graph each part on its domain and indicate the transition at .
The Greatest Integer Function
Definition and Properties
The greatest integer function, denoted , assigns to each real number the greatest integer less than or equal to .
This function is also called the "floor function" and is an example of a step function.
Table: Values of the Greatest Integer Function
x | |
|---|---|
2.8 | 2 |
-1.2 | -2 |
0 | 0 |
3 | 3 |
0.99 | 0 |
-0.25 | -1 |
Example: ; .
