IndietroTypes of Functions and Piecewise-Defined Functions in College Algebra
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2.1 Types of Functions
Families of Functions
In algebra, functions are classified into families based on their algebraic structure and graphical behavior. Understanding these basic types is essential, as they serve as building blocks for more complex functions.
Identity Function:
Squaring Function:
Square Root Function:
Cubing Function:
Cube Root Function:
Reciprocal Function:
Absolute-Value Function:
These fundamental functions are used to model a wide variety of real-world and mathematical phenomena.
Applications of Functions
Modeling with Functions
Functions are used to model relationships between quantities in real-world contexts. For example, the distance between two moving objects can be expressed as a function of time.
Example: Car Distance
Two nurses, Kiara and Matias, drive away from a hospital at right angles to each other.
Kiara's speed: 35 mph; Matias's speed: 40 mph.
Distance between cars after hours: .
Using the Pythagorean theorem, the distance function is:
After 1 hour:
miles 
Piecewise-Defined Functions
Definition and Evaluation
A piecewise-defined function is a function that is defined by different expressions for different intervals of the domain. Only one formula is used for any particular input value, depending on which part of the domain the input belongs to.
Example: 
To evaluate , note that , so use :
Repeat this process for other values by selecting the appropriate formula based on the input.
Graphing Piecewise-Defined Functions
To graph a piecewise-defined function, graph each piece on its specified interval. Use open or closed circles to indicate whether endpoints are included.
Example: 
Graph for and for . Construct tables of values for each piece and plot accordingly.
x < 3 | y = (1/3)x + 3 |
|---|---|
0 | 3 |
1 | 3.33 |
2 | 3.67 |
x ≥ 3 | y = -x |
|---|---|
3 | -3 |
4 | -4 |
More Examples of Piecewise Functions
Piecewise functions can involve linear, quadratic, or other types of expressions.
Example: 
Example: 
Piecewise Functions with Holes and Discontinuities
Some piecewise functions are defined to "patch" holes or discontinuities in otherwise undefined expressions.
Example: 
For , (after simplifying the rational expression). At , the function is defined as , filling in the hole at that point.
Summary Table: Common Piecewise-Defined Functions
Function | Definition | Interval(s) |
|---|---|---|
Absolute Value | ||
Step Function | All real | |
Custom Example | See above |
Additional info: Piecewise-defined functions are widely used in modeling situations where a rule or relationship changes depending on the input value, such as tax brackets, shipping rates, or physical phenomena with thresholds.
