Piecewise Functions Graphing Calculator

Piecewise Function Evaluator

Define up to three pieces of your function below. Then, enter an X value to see the function's output at that point.

Piece 1

and

Piece 2

and

Piece 3

and

Evaluate at X

' + x + '

Piece ' + piece.pieceNum + '

f(x) = ' + piece.funcExpr + '

' + y.toFixed(4) + '

' + x + '

Understanding Piecewise Functions and Their Graphs

A piecewise function is a function defined by multiple sub-functions, each applying to a different interval in the domain. Instead of a single rule, a piecewise function uses different rules for different parts of its input values. This makes them incredibly versatile for modeling real-world situations that change behavior under different conditions.

What is a Piecewise Function?

Imagine a scenario where the cost of a product changes based on the quantity you buy, or the tax rate varies with your income bracket. These are perfect examples of situations that can be modeled using piecewise functions. Each "piece" of the function consists of:

1. A Function Rule: This is the mathematical expression (e.g., f(x) = 2x + 1, f(x) = x^2, f(x) = 5) that defines the output for a specific range of input values.
2. A Domain Interval: This specifies the range of input (x) values for which that particular function rule applies (e.g., x < 0, 0 ≤ x < 5, x ≥ 5).

When you combine these pieces, you get a single function that behaves differently across its domain.

Why Are Piecewise Functions Important?

Piecewise functions are fundamental in various fields:

• Economics: Modeling progressive tax systems, tiered pricing, or supply and demand curves.
• Physics & Engineering: Describing forces that change with distance, electrical signals, or motion with varying acceleration.
• Computer Science: Used in algorithms, data analysis, and defining conditions in programming.
• Everyday Life: Calculating shipping costs based on weight, phone plan charges based on usage, or even speed limits that change on different road segments.

Graphing Piecewise Functions

Graphing a piecewise function involves plotting each sub-function over its specified domain interval. Here's a conceptual approach:

1. Identify Each Piece: Separate the function into its individual rules and their corresponding domains.
2. Graph Each Piece Individually: For each rule, imagine graphing it as a complete function.
3. Apply the Domain Restrictions: Erase or ignore the parts of each graph that fall outside its specified domain interval.
4. Pay Attention to Endpoints:
   • Use an open circle (○) at an endpoint if the interval is strictly less than or greater than (< or >), meaning the point is not included.
   • Use a closed circle (●) at an endpoint if the interval is less than or equal to, or greater than or equal to (≤ or ≥), meaning the point is included.
5. Combine the Segments: The collection of these restricted graphs forms the complete graph of the piecewise function. The graph might have "jumps" or discontinuities where one piece ends and another begins, or it might be continuous if the pieces meet at the same point.

How to Use This Piecewise Function Evaluator

Our calculator allows you to define up to three pieces of a piecewise function and then evaluate the function at a specific X-value. Follow these steps:

1. Define Each Piece:
   • Function f(x) =: Enter the mathematical expression for that piece. Use standard JavaScript math syntax (e.g., x*x for x squared, Math.pow(x, 3) for x cubed, Math.sin(x) for sine of x, Math.abs(x) for absolute value).
   • Lower Bound / Upper Bound: Enter the numerical limits for the domain of this piece. Leave the field empty if there is no bound (e.g., for x < 0, leave "Lower Bound" empty; for x ≥ 2, leave "Upper Bound" empty).
   • Inclusive Checkbox: Check this box if the bound is included in the domain (≤ or ≥); uncheck it if the bound is not included (< or >).
2. Enter X Value: Input the specific numerical value of X at which you want to evaluate the function.
3. Click "Evaluate Function": The calculator will determine which piece's domain the X-value falls into, apply the corresponding function rule, and display the result.

Example Calculation

Let's use the example function defined by default in the calculator:

f(x) =

• x^2, if x < 0
• x, if 0 ≤ x < 2
• 4, if x ≥ 2

Let's evaluate f(1):

1. We check the domains:
   • Is 1 < 0? No. (Piece 1: x < 0)
   • Is 0 ≤ 1 < 2? Yes. (Piece 2: 0 ≤ x < 2)
2. Since 1 falls into the second piece's domain, we use the function f(x) = x.
3. f(1) = 1.

If you enter 1 into the "X Value" field and click "Evaluate Function", the calculator will show f(1) = 1.0000.

Now, let's evaluate f(3):

1. We check the domains:
   • Is 3 < 0? No. (Piece 1: x < 0)
   • Is 0 ≤ 3 < 2? No. (Piece 2: 0 ≤ x < 2)
   • Is 3 ≥ 2? Yes. (Piece 3: x ≥ 2)
2. Since 3 falls into the third piece's domain, we use the function f(x) = 4.
3. f(3) = 4.

If you enter 3 into the "X Value" field and click "Evaluate Function", the calculator will show f(3) = 4.0000.

This calculator is a helpful tool for understanding how different function rules apply across specific intervals, providing a practical way to explore the behavior of piecewise functions.