Understanding Piecewise Functions: Definition, Examples, and Applications

Piecewise function

A piecewise function, also known as a piecewise-defined function, is a mathematical function that is defined by different expressions or formulas for different intervals or subdomains

A piecewise function, also known as a piecewise-defined function, is a mathematical function that is defined by different expressions or formulas for different intervals or subdomains. These intervals or subdomains may have different functions or rules that govern their behavior.

The general form of a piecewise function is:

f(x) = { g1(x) if c1 ≤ x < c2 { g2(x) if c2 ≤ x < c3 { ... { gn(x) if cn ≤ x < cn+1 { ... Here, each gi(x) represents a different function or expression that is defined for a specific interval between two consecutive values of ci. The ci's are commonly referred to as the "breakpoints" or "transition points" where the function switches from one expression to the next. To illustrate this, let's consider an example: f(x) = { x^2 if x < 0 { 2x if 0 ≤ x < 2 { sqrt(3x) if x ≥ 2 In this example, we have a piecewise function with three intervals: 1. For x < 0, the function is defined as f(x) = x^2. This means that if x is less than 0, the output of the function is the square of x. 2. For 0 ≤ x < 2, the function is defined as f(x) = 2x. This means that if x is between 0 (inclusive) and 2 (exclusive), the output of the function is twice the value of x. 3. For x ≥ 2, the function is defined as f(x) = sqrt(3x). This means that if x is greater than or equal to 2, the output of the function is the square root of 3 times x. For example, if we want to evaluate the function at x = -1, we use the first expression: f(-1) = (-1)^2 = 1. If we want to evaluate the function at x = 1, we use the second expression: f(1) = 2(1) = 2. And if we want to evaluate the function at x = 3, we use the third expression: f(3) = sqrt(3(3)) = sqrt(9) = 3. Piecewise functions are often used to represent real-life situations where different formulas apply to different regions or intervals. They can be useful in modeling various phenomena such as rates of change, pricing structures, or any situation that involves distinct rules or behaviors depending on specific conditions.

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