Understanding Piecewise Functions | A Comprehensive Explanation and Examples

piecewise function

A piecewise function is a function that is defined by multiple sub-functions, each defined on a specific interval or region of the input domain

A piecewise function is a function that is defined by multiple sub-functions, each defined on a specific interval or region of the input domain. In other words, a piecewise function consists of different “pieces” or sections, where each piece is defined by a different formula or rule. The function may have different expressions, formulas, or rules that apply to different parts of its domain.

The pieces of a piecewise function are determined by breaking the domain of the function into intervals, and defining a specific formula or rule for each interval. Each formula or rule typically describes the behavior of the function on the corresponding interval.

For example, let’s consider the piecewise function f(x):

f(x) = 2x, if x < 0 x^2, if 0 ≤ x ≤ 3 8, if x > 3

In this case, we have defined three pieces for the function f(x):

1. If x is less than 0, the function is defined as f(x) = 2x. So, if we plug any value less than 0 into the function, it will give us twice that value. For example, f(-2) = 2(-2) = -4.

2. If x is between 0 and 3 (inclusive), the function is defined as f(x) = x^2. So, if we plug any value between 0 and 3 into the function, it will square that value. For example, f(2) = 2^2 = 4.

3. If x is greater than 3, the function is defined as f(x) = 8. So, if we plug any value greater than 3 into the function, it will always give us 8. For example, f(4) = 8.

A piecewise function is useful when there is a need to define a different mathematical rule or expression for different parts of the input domain. Through piecewise functions, we can represent and analyze complex functions that exhibit different behaviors in different regions.

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