Understanding Piecewise Functions in Mathematics

In mathematics, piecewise refers to a function that is defined as the sum of simpler functions, each of which is defined over a specific subdomain or interval of the original function. The term "piecewise" literally means "per piece," and it is used to describe a function that is composed of multiple smaller pieces or subfunctions.

For example, consider the function f(x) = 3x if x < 2, and f(x) = 5x - 2 if x >= 2. This function is defined as the sum of two simpler functions, one for x < 2 and another for x >= 2. The domain of the function is all real numbers, but each subfunction has its own domain: the first subfunction is defined on the interval [0,2) and the second subfunction is defined on the interval (2,∞).

In this case, we say that the function f(x) is piecewise defined as:

f(x) = 3x if x < 2
f(x) = 5x - 2 if x >= 2

The term "piecewise" is often used in calculus and analysis to describe functions that are composed of multiple parts or subfunctions, each of which may have a different domain or range. It is a useful way to break down complex functions into simpler components, making it easier to analyze and understand their properties and behavior.