Piecewise-Defined Functions: Concepts, Graphs, and Applications

Piecewise-Defined Functions

Introduction to Piecewise-Defined Functions

Piecewise-defined functions are mathematical functions defined by different expressions over distinct intervals of the domain. These functions are useful for modeling situations where a rule or relationship changes depending on the input value.

• Definition: A piecewise-defined function is a function composed of two or more sub-functions, each of which applies to a certain interval of the main function's domain.

• Notation: Piecewise functions are typically written using braces to indicate the different cases.

• Example: The absolute value function $f(x) = |x|$ can be written as a piecewise function:

  • $f(x) = x$ if $x \geq 0$

  • $f(x) = -x$ if $x < 0$

Identifying Piecewise Functions from Graphs

Graphs of piecewise-defined functions often show distinct segments, each corresponding to a different rule. To determine the function from a graph, observe the intervals and the behavior of the function in each interval.

• Key Steps:

  1. Identify the intervals on the x-axis where the function changes its rule.

  2. Determine the equation or rule for each segment.

  3. Write the function using piecewise notation.

• Example: If a graph shows a line from $x = -10$ to $x = 2$ and another from $x = 2$ to $x = 8$, write each segment's equation and specify the interval.

Writing Piecewise-Defined Functions

To express a function as piecewise-defined, use the following format:

• General Form: $f(x) = \begin{cases} \text{expression}_1, & \text{if } x \text{ is in interval}_1 \\ \text{expression}_2, & \text{if } x \text{ is in interval}_2 \\ \cdots \end{cases}$

• Example: $f(x) = \begin{cases} 2x, & -10 < x \leq 2 \\ 8 - x, & 2 < x \leq 8 \end{cases}$

Graphing Piecewise-Defined Functions

To graph a piecewise-defined function, plot each segment according to its rule and interval. Pay attention to open and closed endpoints, which indicate whether the endpoint is included in the interval.

• Steps:

  1. For each interval, plot the corresponding expression.

  2. Use solid dots for included endpoints (e.g., $x \leq a$) and open dots for excluded endpoints (e.g., $x < a$).

  3. Check for continuity or jumps at the boundaries.

• Example: For $f(x) = \begin{cases} 2x, & -10 < x \leq 2 \\ 8 - x, & 2 < x \leq 8 \end{cases}$, plot $2x$ from $x = -10$ to $x = 2$, then $8 - x$ from $x = 2$ to $x = 8$.

Step Functions and Applications

Step functions are a type of piecewise-defined function where the output remains constant over intervals. They are commonly used to model pricing, wages, or other situations with fixed rates over ranges.

• Definition: A step function is a function that increases or decreases abruptly at certain points, remaining constant between those points.

• Example: Rental cost for a bike based on hours used:

Hours, x	Cost, y
0 < x ≤ 6	$20
6 < x ≤ 12	$30
12 < x ≤ 18	$40
18 < x ≤ 24	$50

• Domain: $0 < x \leq 24$

• Range: $\{20, 30, 40, 50\}$

• Minimum: $20$

• Maximum: $50$

Modeling Real-World Situations with Piecewise Functions

Piecewise-defined functions are useful for modeling earnings, costs, and other scenarios where the rule changes at certain thresholds.

• Example: Gordon's weekly earnings as a tour guide:

Number of hours worked (h)	Earnings ($)
$0 \leq h \leq 40$	$10h$
$h > 40$	$10(40) + 15(h - 40) = 400 + 15(h - 40)$

• Piecewise Function: $E(h) = \begin{cases} 10h, & 0 \leq h \leq 40 \\ 15h - 200, & h > 40 \end{cases}$

• Application: This function models regular and overtime pay, where the pay rate increases after a threshold.

Summary Table: Types of Piecewise-Defined Functions

Type	Definition	Example
Linear Piecewise	Each segment is a linear function	$f(x) = \begin{cases} 2x, & x < 0 \\ x + 1, & x \geq 0 \end{cases}$
Step Function	Constant value over intervals	Rental cost function above
Absolute Value	Defined by positive and negative cases	$f(x) = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases}$

Key Points for Exam Preparation

• Understand how to read and write piecewise-defined functions from graphs and tables.

• Be able to graph piecewise functions, noting endpoints and intervals.

• Apply piecewise functions to model real-world scenarios, such as wages and costs.

• Know how to determine domain and range from piecewise rules and graphs.

Additional info: These notes expand on the mathematical concepts and applications of piecewise-defined functions, including step functions and modeling scenarios, as inferred from the provided questions and answer keys.