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Piecewise Function Solver

Evaluate and analyze piecewise-defined functions at any x value

Define f(x) with two linear pieces and evaluate at x = p

f(x)={x+, x<

f(x)={x+, x>=2, x=

Piecewise Function Evaluation

1. Compare input x with the boundary value

2. Select the correct piece

3. Substitute x into that expression

4. Check continuity at the boundary

Piecewise functions use different formulas for different input ranges. To evaluate, find which interval x belongs to, then use the corresponding expression. The boundary condition determines which piece applies.

⚠ Check the inequality carefully when the input equals the boundary value. The condition (x<a vs x>=a) determines which piece applies at the boundary.

What Is a Piecewise Function?

A piecewise function is defined by multiple sub-functions, each applying to a specific interval. Common examples include absolute value |x| = {x if x>=0, -x if x<0}, tax brackets, and step functions. Evaluation requires checking conditions first.

Branch Selection

Check which interval x falls into. Use the corresponding expression. Only one piece is active for any given x.

Boundary Points

At x=boundary, check which inequality applies. The < vs <= determines which piece to use at the exact boundary.

Continuity Check

A piecewise function is continuous at the boundary if both pieces give the same value when approaching from left and right.

Absolute Value

The classic piecewise: |x| = {x if x>=0, -x if x<0}. This function is continuous but not differentiable at x=0.

Teaching Example: f(x) = {2x if x<2, 3x-2 if x>=2}, evaluate at x=3. Since 3>=2, use second: f(3)=3*3-2=9-2=7. Check continuity at x=2: left=4, right=4, continuous!

Applications

Algebra Tax Brackets Signal Processing Computer Science Economics

Frequently Asked Questions

What is piecewise?▼

Function defined by different expressions on different intervals. Check x range, select correct expression, evaluate.

How to evaluate at boundary?▼

Check the inequality sign. x<2 uses first piece, x>=2 uses second. At x=2 exactly, the >= condition selects the second piece.

Continuity at boundary?▼

If left and right limits equal the function value at the boundary, the function is continuous there. Otherwise it has a jump.

Real-world uses?▼

Tax brackets, shipping costs, absolute value, step functions in signal processing, conditional discounts.

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