Graph a Piecewise Function Calculator

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Expert Verified by: David Chen, CFA | Senior Financial Analyst & Mathematics Consultant

Master complex functions effortlessly with our graph a piecewise function calculator. This tool helps you define multiple function segments over specific intervals, evaluate precise values, and visualize the transition points instantly.

Graph a Piecewise Function Calculator

Result for $f(x)$:
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Graph a Piecewise Function Calculator Formula

$$f(x) = \begin{cases} m_1x + b_1 & \text{if } x < k_1 \\ m_2x + b_2 & \text{if } k_1 \le x < k_2 \\ m_3x + b_3 & \text{if } x \ge k_2 \end{cases}$$

Reference: Khan Academy – Piecewise Functions | Wolfram MathWorld

Variables:

  • $m$ (Slope): The rate of change for that specific segment.
  • $b$ (Y-intercept): The value of the segment when $x = 0$ (if applicable).
  • $k$ (Boundary Points): The critical values of $x$ where the function changes its definition.
  • $x$: The independent variable used for evaluation and plotting.

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What is a Piecewise Function?

A piecewise function is a function defined by multiple sub-functions, where each sub-function applies to a certain interval of the main function’s domain. In mathematics, this is crucial for modeling real-world scenarios like tax brackets, shipping costs, or physical forces that change abruptly.

Graphing a piecewise function involves plotting each “piece” only within its designated $x$ range. Our graph a piecewise function calculator automates this process by checking boundaries and rendering a visual representation instantly.

How to Calculate a Piecewise Function (Example)

  1. Identify the $x$ value you want to evaluate.
  2. Compare $x$ against the boundary values ($k_1, k_2$).
  3. Select the sub-function corresponding to the correct interval.
  4. Plug $x$ into that specific linear equation ($mx + b$).
  5. Verify if there are any points of discontinuity at the boundaries.

Frequently Asked Questions (FAQ)

Are piecewise functions continuous? Not necessarily. A piecewise function is continuous only if the pieces meet at the boundary points (i.e., the limits from the left and right are equal).

Can I have more than three pieces? While this calculator focuses on three pieces for clarity, the mathematical concept allows for an infinite number of sub-functions.

What is a jump discontinuity? This occurs when the pieces do not “link up” at the boundary, creating a vertical gap in the graph.

How do I find the domain? The domain is the union of all defined intervals for the sub-functions.

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