Lesson Objectives

- Learn how to identify the absolute value function
- Learn how to identify the greatest integer function
- Learn how to evaluate a piecewise-defined function
- Learn how to graph a piecewise-defined function

## How to Graph and Evaluate a Piecewise-Defined Function

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.

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 greater than 2, f(x) or the function's value is just -6. $$f(3)=-6$$

Example #2: Sketch the graph of each function. $$\ f(x)=\begin{cases}(x + 4)^2 & \text{if}\hspace{.2em}x ≤ -3 \\ 2x + 4 & \text{if}\hspace{.2em}x > -3 \end{cases}$$ To get the graph of f(x) = (x + 4)

### The Absolute Value Function

The absolute value function pairs every real number with its absolute value. $$f(x)=|x|$$ $$\text{domain}:(-\infty, \infty)$$ $$\text{range}: [0, \infty)$$ This function is continuous over its entire domain. The absolute value function is an example of a piecewise-defined function. It has different rules across different intervals of the domain. For x-values that are non-negative (0 or some positive real number), we just get the number. In the case where x-values are negative, we get the opposite of the number. If you look at the graph above, we have the line y = x to the right of the y-axis and the line y = -x to the left of the y-axis. $$\ f(x)=|x|=\begin{cases}x & \text{if}\hspace{.2em}x ≥ 0 \\ -x & \text{if}\hspace{.2em}x < 0 \end{cases}$$### The Greatest Integer Function

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. The greatest integer function is an example of a step function. $$f(x)=[x]$$ $$\text{domain}: (-\infty, \infty)$$ $$\text{range}: \{y | y ∈ \mathbb{Z}\}$$ The Z above is used for the set of integers. In other words, y can be any integer. The greatest integer function is also commonly known as the floor function. The notation for this function can vary quite a bit. You may also see this written as: $$f(x)=\lfloor x \rfloor$$ $$f(x)=[\![x]\!]$$ This function is discontinuous at all integer values. $$\text{for}\hspace{.2em}{-}2 ≤ x < -1, f(x)=-2$$ $$\text{for}\hspace{.2em}{-}1 ≤ x < 0, f(x)=-1$$ $$\text{for}\hspace{.2em}0 ≤ x < 1, f(x)=0$$ $$\text{for}\hspace{.2em}1 ≤ x < 2, f(x)=1$$ $$\text{for}\hspace{.2em}2 ≤ x < 3, f(x)=2$$ Note: The dots in purple are used to indicate that the graph continues indefinitely in the same pattern.### 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 greater than 2, f(x) or the function's value is just -6. $$f(3)=-6$$

### Graphing a Piecewise-Defined Function

Additionally, we may be asked to graph a piecewise-defined function. Let's look at an example.Example #2: Sketch the graph of each function. $$\ f(x)=\begin{cases}(x + 4)^2 & \text{if}\hspace{.2em}x ≤ -3 \\ 2x + 4 & \text{if}\hspace{.2em}x > -3 \end{cases}$$ To get the graph of f(x) = (x + 4)

^{2}, we can grab a few points such as: $$(-7, 9)$$ $$(-6, 4)$$ $$(-4, 0)$$ $$(-3, 1)$$ To get the graph of f(x) = 2x + 4, we can use the y-intercept: $$(0, 4)$$ And then use the slope of 2 to get additional points. One additional point would be: $$(2, 8)$$#### Skills Check:

Example #1

Find f(-1) $$\ f(x)=\begin{cases}x^2 & \text{if}\hspace{.2em}x ≤ 0 \\ 2x - 4 & \text{if}\hspace{.2em}x > 0 \end{cases}$$

Please choose the best answer.

A

$$f(-1)=-6$$

B

$$f(-1)=1$$

C

$$f(-1)=-1$$

D

$$f(-1)=3$$

E

$$f(-1)=2$$

Example #2

Find f(3) $$\ f(x)=\begin{cases}x - 1 & \text{if}\hspace{.2em}x ≤ -3 \\ x + 4 & \text{if}\hspace{.2em}x > -3 \end{cases}$$

Please choose the best answer.

A

$$f(3)=7$$

B

$$f(3)=2$$

C

$$f(3)=-4$$

D

$$f(3)=1$$

E

$$f(3)=9$$

Example #3

Find f(-2) $$\ f(x)=\begin{cases}x^2 - 3 & \text{if}\hspace{.2em}x ≤ 3 \\ (x - 3)^2 & \text{if}\hspace{.2em}x > 3 \end{cases}$$

Please choose the best answer.

A

$$f(-2)=25$$

B

$$f(-2)=13$$

C

$$f(-2)=9$$

D

$$f(-2)=1$$

E

$$f(-2)=8$$

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