Piecewise

In mathematics, computer science, and related fields, 'piecewise' describes something (a function, a curve, a definition, etc.) that is defined by multiple sub-definitions or expressions, each applicable over a specific interval or condition within the overall domain. Essentially, it's constructing something complex by combining simpler, distinct components. The behavior or value of the piecewise construct varies depending on the input, guided by the applicable rule based on the input's characteristics.

Piecewise meaning with examples

• In signal processing, a piecewise linear function might model the amplitude of a sound wave, defined differently across distinct time segments. For example, a rapid rise in amplitude over the first second, a plateau for the next two, then a gradual decline, each requiring its own separate linear equation.
• Consider a piecewise function describing shipping costs: free for orders under $50, a flat fee for orders between $50 and $100, and a higher flat fee for orders above $100. The final shipping cost calculation is based on the order's value, which determines which piece is used.
• A piecewise defined curve in computer graphics could represent a complex path. Each segment could be a Bezier curve or a simple line, but collectively, they form a more intricate and nuanced visual structure. The appropriate segment is selected depending on the input value (e.g., a point along the curve).
• When calculating taxes, the tax rate might be applied piecewise. Different income brackets are taxed at different rates. The overall tax liability is the sum of taxes calculated at each bracket, representing a combination of linear functions acting on different intervals of income.