Piecewise functions

Piecewise functions are mathematical functions defined by multiple sub-functions, each applicable to specific intervals or conditions of the independent variable. These functions can take different forms depending on the input value, making them useful for modeling scenarios where behavior changes at certain thresholds.

Key Concepts:

Definition: A piecewise function is expressed using different formulas for different parts of its domain. It is typically written in a form like:
$f(x) = \begin{cases} f_1(x) & \text{if } x < a \\ f_2(x) & \text{if } a \leq x < b \\ f_3(x) & \text{if } x \geq b \end{cases}$
Domain and Range: Each sub-function has its own domain, and the overall function's domain is the union of these intervals. The range depends on the outputs of the individual functions.
Continuity: Piecewise functions can be continuous or discontinuous, depending on how the sub-functions connect at their boundaries.
Applications: These functions are commonly used in real-world scenarios, such as tax brackets, shipping costs, and certain physical phenomena.
Graphing: When graphing piecewise functions, each segment is plotted according to its specific rule over its defined interval, often resulting in a jagged or segmented graph.

Overall, piecewise functions are a powerful tool for describing complex behaviors in a clear and systematic way.

In this article

Part 1: Introduction to piecewise functions Part 2: Worked example: evaluating piecewise functions Part 3: Worked example: graphing piecewise functions Part 4: Worked example: domain & range of step function Part 5: Worked example: domain & range of piecewise linear functions

Part 1: Introduction to piecewise functions

A piecewise function is a function built from pieces of different functions over different intervals. For example, we can make a piecewise function f(x) where f(x) = -9 when -9 < x ≤ -5, f(x) = 6 when -5 < x ≤ -1, and f(x) = -7 when -1

Here are the key points to learn when studying "Introduction to Piecewise Functions":

Definition: Understand that a piecewise function is defined by different expressions based on different intervals of the input variable.
Notation: Familiarize yourself with the standard notation, which includes specifying the function's conditions clearly for each piece.
Graphing: Learn how to graph piecewise functions by plotting each section based on its defined interval, ensuring to consider whether the endpoints are included (open or closed circles).
Domain and Range: Determine the domain by identifying all possible x-values as per the function's intervals, and analyze the range based on the outputs of each piece.
Continuity: Understand concepts of continuity and discontinuity at the boundaries of the pieces, and how to determine if the function is continuous at those points.
Evaluating: Practice evaluating the function at various points to reinforce understanding of which expression to use based on the input value.
Applications: Recognize the practical applications of piecewise functions in real-world problems, such as tax brackets or shipping rates.

By mastering these points, you will have a solid foundation in understanding piecewise functions.

Part 2: Worked example: evaluating piecewise functions

A piecewise function is a function that is defined in separate "pieces" or intervals. For each region or interval, the function may have a different equation or rule that describes it. We can evaluate piecewise functions (find the value of the function) by using their formulas or their graphs.

When studying "evaluating piecewise functions," focus on these key points:

Definition: A piecewise function is defined by multiple sub-functions, each applicable to a specific interval of the input variable.
Identifying Intervals: Pay attention to the conditions that dictate which sub-function to use based on the value of the input (often denoted as $x$ ).
Substitution: To evaluate the function at a certain point, determine which piece of the function corresponds to that point and use it to perform the calculation.
Boundary Values: Be mindful of boundary values; sometimes a value may lie exactly on the boundary of two intervals, and it’s essential to know which function to use in such cases.
Graphing: Visualizing piecewise functions can help in understanding how the function behaves across different intervals.
Continuous vs. Discontinuous: Determine if the piecewise function is continuous or has jumps at the boundaries, which can affect the function's behavior and evaluation.
Practice: Work through various examples to reinforce understanding of how to evaluate piecewise functions effectively.

By mastering these points, you will be well-equipped to understand and evaluate piecewise functions.

Part 3: Worked example: graphing piecewise functions

A piecewise function is a function that is defined in separate "pieces" or intervals. For each region or interval, the function may have a different equation or rule that describes it. We can graph a piecewise function by graphing each individual piece.

When studying "Worked Example: Graphing Piecewise Functions," focus on these key points:

Definition of Piecewise Functions: Understand that piecewise functions are defined by different expressions based on specific intervals of the input variable.
Identifying Intervals: Learn to identify the intervals for each piece of the function. These are specified by inequalities or conditions in the function’s definition.
Evaluating Each Piece: For each interval, substitute values to evaluate the corresponding expression. This helps in determining key points on the graph.
Plotting Points: Plot the key points obtained from evaluating the function within the defined intervals on a coordinate plane.
Connecting Points: Connect the points for each piece with the appropriate type of line (solid or dashed) to indicate whether the endpoints are included or excluded.
Continuity: Check for continuity at the transition points between pieces to determine if the graph is continuous or has any jumps.
Practice with Examples: Work through multiple examples to solidify your understanding of how to graph different types of piecewise functions.

By focusing on these points, you'll gain a solid understanding of how to graph piecewise functions effectively.

Part 4: Worked example: domain & range of step function

Finding the domain and range of a piecewise function that is constant in each segment. Such functions are called "step functions."

When studying the domain and range of a step function, focus on the following key points:

Definition of Step Function: Understand that a step function is a piecewise constant function, which means its values remain constant within specified intervals.
Domain: Identify the domain, which is typically all real numbers or a specific set of values where the function is defined. For a step function, it usually covers a range of intervals.
Range: Determine the range by identifying the distinct output values the function takes. These values correspond to the function's constant pieces.
Graphing Step Functions: Recognize that the graph of a step function consists of horizontal segments (steps) at constant levels, with vertical jumps at the boundaries of the intervals.
Interval Notation: Use interval notation to express the domain and range clearly. For example, the domain may be expressed as (-∞, ∞), while the range consists of a set of distinct values.
Example Analysis: Work through examples to see how the specific values and intervals in a step function define its domain and range.

By focusing on these points, you can effectively understand the characteristics and behaviors of step functions in terms of their domain and range.

Part 5: Worked example: domain & range of piecewise linear functions

Finding the domain and range of a piecewise function where each segment is linear.

When studying "Worked example: domain & range of piecewise linear functions," focus on the following key points:

Understanding Piecewise Functions: Recognize that a piecewise function has different expressions for different parts of its domain.
Identifying Domain: Determine the domain by identifying the intervals of x-values for which the function is defined. This includes finding any restrictions based on the piecewise segments.
Finding Range: Analyze the outputs of the function for each segment to determine the range. This involves evaluating the function at critical points (e.g., endpoints and points of intersection) to find the minimum and maximum values.
Graphical Representation: Learn to sketch the graph of piecewise functions, highlighting each linear section and marking any discontinuities.
Continuity and Discontinuity: Understand when the function is continuous or discontinuous at the boundaries of the piecewise segments.
Example Workthroughs: Practice with concrete examples to reinforce the process of extracting domain and range, connecting algebra with graphical interpretation.

By mastering these concepts, you can effectively analyze piecewise linear functions and their characteristics.