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Chapter 2: Problem 123

What is a piecewise function?

Short Answer

Expert verified
A piecewise function is a type of function that is defined by several different formulas or 'pieces', each of which applies to a different domain.

Step by step solution

01

Definition

A piecewise function is a function defined by several different formulas or 'pieces', each of which applies to a different domain. The function changes according to which part of the domain you are on.
02

Structure of a Piecewise Function

A piecewise function is usually represented as: \(f(x) = \begin{cases} f_1(x) & \text{if } x \in D_1 \\ f_2(x) & \text{if } x \in D_2 \\ \vdots & \vdots \\ f_n(x) & \text{if } x \in D_n \end{cases}\) where each \(f_i(x)\) is a function and each \(D_i\) is a subset of the total domain of \(f(x)\). The \(D_i\) sets are usually mutually exclusive (one number in the domain only belongs to one set) and collectively exhaustive (all the numbers in the domain belong to one of the sets).
03

Example of a Piecewise Function

An example of a piecewise function is the absolute value function, \(f(x)=|x|\), which can be defined as: \[f(x) = \begin{cases} -x & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases}\]

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