Problem 104 - College Algebra Essentials

Question

What is a piecewise function?

Step-by-Step Solution

Answer

A piecewise function is a function that is defined by several different sub-functions, each of which applies to a different interval on the number line. For example, the function could look like \(f(x) = x^2\) for \(x < 0\) and \(f(x) = 2x\) for \(x \geq 0\).

Step 1: Definition of Piecewise Function

A piecewise function is a type of function that is defined by multiple sub-functions, each applying to a different interval in the domain of the function. It is called 'piecewise' because it is defined by different pieces or sections of various functions.

Step 2: Understanding Piecewise Function

A piecewise function is often depicted graphically with a number line indicating the distinct intervals that apply for each sub-function. These intervals are usually separated by a single point, at which the function value may be either one of the sub-function values or a completely separate value.

Step 3: Example of Piecewise Function

For example, consider the following piecewise function f(x): \[ f(x) = \begin{cases} x^2, & \text{if } x < 0 \ 2x, & \text{if } x \geq 0 \end{cases}\] This function takes the form of the square function \(x^2\) for values of \(x\) less than 0, and the form of the linear function \(2x\) for values of \(x\) equal to or greater than 0.