Problem 104 - College Algebra

Question

What is a piecewise function?

Step-by-Step Solution

Answer

A piecewise function is a function defined by different formulas for different parts of its domain. The formula used depends on the input value. An example is the absolute value function.

Step 1: Definition

A piecewise function is a function that is defined by several different formulas or 'pieces', each applying to a different section of its domain.

Step 2: Operation

The specific formula used to calculate a piecewise function's value depends on the value of its input. The domain of the function is divided into sections, and a different formula is used for each section. The formulas are usually chosen so that the function is continuous, meaning that it doesn't have any abrupt changes or jumps.

Step 3: Example

A simple example of a piecewise function is the absolute value function. This function is defined by two different formulas: For \( x \) greater than or equal to 0, the function is equal to \( x \); and for \( x \) less than 0, the function is equal to -\( x \). So, the absolute value function can be written as: \( f(x) = \begin{cases} x & \text{if } x\geq 0\\ -x & \text{if } x < 0 \end{cases} \).