Enter Absolute Value Function
Use vertical bars for absolute values and variable x.
Interiors must be linear, for example: |2x-3| + |x+1| - 4.
Multiple absolute terms are supported.
Formulas Used
For each linear interior ax + b: |ax + b| = ax + b when ax + b ≥ 0, and |ax + b| = -(ax + b) when ax + b < 0. Breakpoints at ax + b = 0 split the real line into intervals.
How to Use This Calculator
- Enter your function using x and | |, e.g. |2x-3| + |x+1| - 4.
- Optionally set custom minimum, maximum, and step size.
- Optionally choose a specific x-value to evaluate f(x).
- Click “Calculate Piecewise Form”.
- Review piecewise form, breakpoints, sample table, and preview plot.
- Export CSV, PDF, or TXT for further use.
Worked Example: |2x - 3| + |x + 1| - 4
Breakpoints at x = -1 and x = 1.5 split the domain. On each interval the calculator chooses appropriate signs, producing explicit linear pieces plus sample values for verification.
Key Features of the Piecewise Function Engine
Detects absolute interiors, computes breakpoints, builds intervals, and outputs explicit linear pieces. Supports multiple terms, configurable sampling, one-point evaluation, quick visualization, and text/CSV/PDF exports for rigorous analysis.
Use Cases for Absolute Value Piecewise Analysis
Apply it to distance models, tolerance bands, penalty costs, break-even regions, exam preparation, and validating algebraic work in technical, financial, or educational contexts involving absolute value expressions.
Understanding Breakpoints and Graph Behavior
Each breakpoint corresponds to an interior equal to zero. The tool highlights where slopes change, exposing corners and non-differentiable points and clarifying continuity on each interval.
Benefits for Teachers, Students, and Analysts
Ideal for live demonstrations, homework checking, structured solutions, and model documentation. Ensures consistent, transparent handling of absolute values across tasks.
FAQs
1) What expressions are supported?
Expressions with one variable x, where each absolute interior is linear, like ax+b. Multiple absolute terms, basic arithmetic, and parentheses are supported. Non-linear interiors and other functions are not processed.
2) How are breakpoints and intervals determined?
The tool finds each interior ax+b, solves ax+b=0, and uses those roots as breakpoints. They partition the real line, and each interval receives the correct linear pieces.
3) Why do some rows show “NaN”?
“NaN” appears when the expression cannot be evaluated at that x, often due to invalid syntax, missing bars, or extreme values. Check formatting, keep interiors linear, and adjust the sample range.
4) Can I use nested or non-linear absolute values?
This version supports linear interiors only. Nested or non-linear absolute expressions are not expanded automatically. For such cases, analyze sign changes manually before forming your own piecewise function.
5) How can I export and reuse the results?
Use Download CSV for data, Download PDF for a table snapshot, and Download Piecewise Definition (TXT) for the final form. These files can be shared, graphed, or embedded into reports.