Piecewise Function Calculator - Evaluate f(x)

Piecewise Function Calculator

Use 'x'. Math functions like pow(), sin() okay.
Boolean expression involving 'x'. Use ==, !=, <, <=, >, >=, && (AND), || (OR).

Result:

f() =

What is a Piecewise Function?

A piecewise-defined function (or simply piecewise function) is a function defined by multiple sub-functions, where each sub-function applies to a specific interval or condition within the domain. Instead of a single formula defining the output, the formula used depends on the value of the input x.

The standard notation looks like this:

f(x) = { Expression 1 , if Condition 1 is true { Expression 2 , if Condition 2 is true { ... { Expression k , if Condition k is true

To evaluate a piecewise function at a given value of x, you first determine which condition Condition i is satisfied by x. Then, you use the corresponding Expression i to calculate the function's output value f(x).

These functions are useful for modeling situations that change behavior under different circumstances, like tax brackets, shipping costs based on weight, or physical phenomena with different states.

How This Piecewise Function Calculator Works

This calculator evaluates a piecewise function that you define for a specific input value x.

  1. Define Pieces:
    • For each piece of the function, enter the mathematical Expression in terms of 'x'. You can use standard operators (+ - * / %) and JavaScript's Math functions (pow(), sin(), cos(), exp(), log(), sqrt(), abs(), PI, etc.).
    • Enter the corresponding boolean Condition that determines when that expression applies. Use 'x' and comparison operators (< <= > >= == !=) and logical operators (&& for AND, || for OR). Example: x >= 0 && x < 10.
    • Use the "+ Add Another Piece" button to define as many pieces as needed. Use the "X" button to remove unwanted pieces.
  2. Input x-Value: Enter the specific numerical value of x at which you want to evaluate the function.
  3. Click Evaluate: Press the "Evaluate f(x)" button.
  4. Condition Matching: The calculator iterates through the defined pieces in the order you entered them. For each piece, it evaluates its Condition using the input x value.
  5. Function Evaluation: As soon as it finds the first piece whose Condition evaluates to true, it stops searching and evaluates the corresponding Expression for that piece using the input x value.
  6. Display Result: The calculator shows the calculated output value f(x) and indicates which piece's condition was met. If no conditions are met by the input x, it displays an error message.
Security Note: This calculator uses JavaScript's new Function() constructor (or similar techniques) to evaluate the user-provided expressions and conditions. While flexible, executing arbitrary code from user input can be a security risk in different contexts. Use this tool responsibly and be cautious about the code you enter.

Frequently Asked Questions (FAQs)

Q1: How do I write the expressions and conditions?

Use standard JavaScript syntax:

  • Variable: Use 'x' (case-sensitive).
  • Operators: + - * / % (modulo), ** or pow(x,y) for exponentiation.
  • Comparisons: < <= > >= == (equal to), != (not equal to).
  • Logical: && (AND), || (OR), ! (NOT). Use parentheses () to group conditions.
  • Math Functions: Use JavaScript's Math object, e.g., Math.sin(x), Math.sqrt(x), Math.abs(x), Math.exp(x), Math.PI. Note: The calculator internally tries to allow using sin(x) etc. directly, but prefixing with `Math.` is safer.
  • Numbers: Integers (5) and decimals (3.14) are fine.

Condition Example: To represent 0 ≤ x < 10, write x >= 0 && x < 10.

Expression Example: To represent 3x² + π, write 3 * pow(x, 2) + PI or 3*x*x + Math.PI.

Q2: What happens if my input 'x' satisfies multiple conditions?

The calculator evaluates the pieces in the order they appear on the screen. It uses the expression corresponding to the first condition that evaluates to true for the given x value and then stops searching. Make sure your conditions correctly partition the domain as intended.

Q3: What happens if my input 'x' does not satisfy any condition?

If the input x does not make any of the provided conditions true, the function is technically undefined at that point according to your definition. The calculator will display an error message indicating that the value does not fall into any defined domain/piece.

Q4: Is there a limit to the number of pieces I can define?

There's no hard limit enforced by the calculator itself, but performance might degrade slightly with a very large number of pieces. Browser memory or performance limitations might eventually become a factor for an extremely high number of pieces. Keep it reasonable for practical use.

Q5: Can I define conditions like "x is an integer"?

Yes, you can use JavaScript expressions. For example, to check if x is an integer, you could use the condition Number.isInteger(x) or x % 1 == 0.

Q6: What if my expression or condition has a syntax error?

The calculator uses JavaScript's evaluation mechanisms. If you enter an expression or condition with incorrect syntax (e.g., unmatched parentheses, invalid operators, undefined variables other than 'x'), the evaluation will likely fail, and the calculator will display an error message indicating a problem with that specific piece. Double-check your syntax carefully.