Piecewise Function Graphing Calculator
Define Your Piecewise Function
Enter your function segments and their corresponding domains. The calculator will evaluate and represent the function's behavior.
Note: Use 'x' as the variable. Supports basic arithmetic (+, -, *, /), powers (^), and common functions like sin(x), cos(x), tan(x), log(x), exp(x), sqrt(x). Ensure domains do not overlap unexpectedly and cover all desired values of x.
Function Evaluation
Enter function definitions and domains to see evaluated points.
Understanding Piecewise Functions
A piecewise function is a function defined by multiple sub-functions, each applying to a certain interval of the main function's domain. Think of it as different rules for different parts of the input values.
Mathematical Definition
A piecewise function can be represented mathematically as:
f(x) = { f1(x), if x ∈ [a1, b1)
{ f2(x), if x ∈ [a2, b2)
{ f3(x), if x ∈ [a3, b3]
...
where:
f(x)is the overall piecewise function.fi(x)are the individual sub-functions (or "pieces").[ai, bi)or[ai, bi](and other interval notations) define the specific domain or interval over which each sub-functionfi(x)is valid.
Why Graph Piecewise Functions?
Graphing piecewise functions is crucial for visualizing their behavior across different intervals. It helps in understanding:
- Continuity and Discontinuity: Where the graph "jumps" or has breaks.
- Overall Shape: How the function behaves from negative infinity to positive infinity.
- Specific Values: The exact output for a given input.
- Real-World Applications: Modeling scenarios with changing conditions, such as tax brackets, utility pricing, or complex physical phenomena.
How This Calculator Works
This calculator allows you to input up to three different function segments and their respective domain intervals. For each segment, you provide:
- The Function Expression: This is the mathematical formula for that part of the function (e.g.,
2*x + 1,x^2,sin(x)). - The Domain Interval: This specifies the range of 'x' values for which the function expression is applied. You define the start (minimum) and end (maximum) values of this interval.
After you input these details, the calculator evaluates the function at several points within each defined domain to give you a textual representation of the function's behavior. For a visual representation, you would typically use a dedicated graphing tool or library.
Example Usage
Consider the following piecewise function:
g(x) = { x + 2, if x < 0
{ x^2, if 0 ≤ x < 2
{ 4, if x ≥ 2
To input this into the calculator:
- Function 1:
x + 2, Domain:-10to-0.001(approximating x < 0) - Function 2:
x^2, Domain:0to1.999(approximating 0 ≤ x < 2) - Function 3:
4, Domain:2to10(approximating x ≥ 2)
The calculator would then evaluate these segments. For instance, for the first segment, it might show points like (-10, -8), (-5, -3), (-1, 1). For the second, (0, 0), (1, 1), (1.5, 2.25), (1.999, 3.996). For the third, (2, 4), (5, 4), (10, 4).
Evaluated Points:
- ";
var pointsPerDomain = 5; // Number of points to evaluate in each domain segment
// Validate domains and functions before proceeding
var isValid = true;
if (isNaN(domain1Start) || isNaN(domain1End) || domain1Start >= domain1End) isValid = false;
if (isNaN(domain2Start) || isNaN(domain2End) || domain2Start >= domain2End) isValid = false;
if (isNaN(domain3Start) || isNaN(domain3End) || domain3Start >= domain3End) isValid = false;
if (func1 === "" && func2 === "" && func3 === "") {
outputHTML += "
- Please enter at least one function and domain. "; isValid = false; } if (!isValid) { resultDiv.innerHTML = "Error: Please ensure all domain start values are less than their corresponding end values and are valid numbers. You must also enter at least one function."; return; } // Evaluate Function 1 if (func1 && !isNaN(domain1Start) && !isNaN(domain1End)) { var step1 = (domain1End – domain1Start) / (pointsPerDomain – 1); for (var i = 0; i < pointsPerDomain; i++) { var x = domain1Start + i * step1; var y = safeEval(func1, x); if (!isNaN(y)) { outputHTML += "
- f(" + x.toFixed(2) + ") = " + y.toFixed(2) + " (using '" + func1 + "') "; } } // Add endpoint if not already covered and domain is finite if (!isNaN(domain1End)) { var yEnd = safeEval(func1, domain1End); if (!isNaN(yEnd)) { outputHTML += "
- f(" + domain1End.toFixed(2) + ") = " + yEnd.toFixed(2) + " (using '" + func1 + "') "; } } } // Evaluate Function 2 if (func2 && !isNaN(domain2Start) && !isNaN(domain2End)) { var step2 = (domain2End – domain2Start) / (pointsPerDomain – 1); for (var i = 0; i < pointsPerDomain; i++) { var x = domain2Start + i * step2; var y = safeEval(func2, x); if (!isNaN(y)) { outputHTML += "
- f(" + x.toFixed(2) + ") = " + y.toFixed(2) + " (using '" + func2 + "') "; } } // Add endpoint if not already covered and domain is finite if (!isNaN(domain2End)) { var yEnd = safeEval(func2, domain2End); if (!isNaN(yEnd)) { outputHTML += "
- f(" + domain2End.toFixed(2) + ") = " + yEnd.toFixed(2) + " (using '" + func2 + "') "; } } } // Evaluate Function 3 if (func3 && !isNaN(domain3Start) && !isNaN(domain3End)) { var step3 = (domain3End – domain3Start) / (pointsPerDomain – 1); for (var i = 0; i < pointsPerDomain; i++) { var x = domain3Start + i * step3; var y = safeEval(func3, x); if (!isNaN(y)) { outputHTML += "
- f(" + x.toFixed(2) + ") = " + y.toFixed(2) + " (using '" + func3 + "') "; } } // Add endpoint if not already covered and domain is finite if (!isNaN(domain3End)) { var yEnd = safeEval(func3, domain3End); if (!isNaN(yEnd)) { outputHTML += "
- f(" + domain3End.toFixed(2) + ") = " + yEnd.toFixed(2) + " (using '" + func3 + "') "; } } } if (outputHTML === "
- No valid points could be calculated. Check your functions and domains. "; } outputHTML += "
Evaluated Points:
- ") {
outputHTML += "