Piecewise-Defined Functions Lesson

In this lesson, we will learn about the piecewise-defined function, which is also known as the split-definition function. This type of function is defined by different rules over different intervals of the domain. Let's begin with the absolute value function.

The Absolute Value Function

$$f(x)=|x|$$ $$domain:(-\infty, \infty)$$ $$range : [0, \infty)$$ The absolute value function is an example of a piecewise-defined function. It has different rules across different intervals of the domain. $$\ f(x)=|x|=\begin{cases}x, & \text{if}\hspace{.2em}x ≥ 0 \\ -x & \text{if}\hspace{.2em}x < 0 \end{cases}$$

The Greatest Integer Function

$$f(x)=[x]$$ $$domain: (-\infty, \infty)$$ $$range: \{y | y ∈ \mathbb{Z}\}$$ The greatest integer function is another example of a piecewise-defined function. This function pairs every real number x with the greatest integer that is less than or equal to x.

Evaluating a Piecewise-Defined Function

In some cases, we may be asked to evaluate a piecewise-defined function for a given value of the domain. Let's look at an example.
Example #1: Find f(-1) and f(3). $$\ f(x)=\begin{cases}4 - x^3, & \text{if}\hspace{.2em}x ≤ 2 \\ -6 & \text{if}\hspace{.2em}x > 2 \end{cases}$$ Let's begin with f(-1). To find f(-1), we think about the rule associated with an x-value of -1. $$\text{if}\hspace{.2em}x ≤ 2 :$$ $$f(x)=4 - x^3$$ So we want to plug in a -1 for x here: $$f(-1)=4 - (-1)^3$$ $$f(-1)=4 + 1=5$$ $$f(-1)=5$$ Our second task is to find f(3). Here, the process is a bit easier. When x is larger than 2, f(x) or the function's value is just -6. $$f(3)=-6$$

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