## High School Algebra I Unlocked (2016)

### Chapter 7. Function Basics

### Lesson 7.2. Intervals and Interval Notation

Often the domain and range of a function will be expressed in **interval notation**. An **interval** is a set of real numbers between, and at times including, two numbers. Consider the following examples that express the domain of a function using interval notation:

The use of brackets (**[ ]**) in interval notation indicates that the value of the endpoint is included in the interval; graphically, the endpoints will be filled circles. Conversely, the use of parentheses (**( )**) in interval notation indicates that the value of the endpoint is not included in the interval; graphically, the endpoints will be unfilled circles.

Consider the graph of *g*(*x*). In this function, the least *x*-value is −4, the greatest *x*-value is 3, and all of the points on the graph are depicted as filled circles. Therefore, the domain of *g*(*x*) includes all real numbers between −4 and 3, including both −4 and 3, and is written in interval notation as [−4, 3]. You can also use interval notation to express the range, or all possible *y*-values, of a function. Let’s try a couple questions that focus on interval notation.

**EXAMPLE **

**Express the range of g(x) in interval notation.**

In the graph of *g*(*x*), the lowest *y*−value is −3, which occurs at point (−4, −3), and the greatest *y*-value is 2, which occurs at point (1, 2). Since both of these points are filled-in circles, and therefore included in the range, the function *g*(*x*) has a range of [−3, 2].

**EXAMPLE **

**Express the domain and range of h(x) in interval notation.**

Since *h*(*x*) has *x*-values that range from −4 to 3, but point (−4, −3) has an unfilled endpoint, the domain of *h*(*x*) is (−4, 3]. Similarly, *h*(*x*) has *y*-values that range from −3 to 2, but point (−4, −3) has an unfilled endpoint, signifying that the range of *h*(*x*) is (−3, 2].

Thus, the function *h*(*x*) has a domain of (−4, 3] and a range of (−3, 2].

But what if the function is discontinuous at various points? How would it be represented in interval notation? To express a domain that is true over multiple intervals, you use the **union** symbol: ∪. To understand how this works, look at the graph below.

Let’s use this graph to determine the domain and range of *m*(*x*), and state them in interval notation. Based on the graph, you can determine that *m*(*x*) is has *x*-values that range from *x* = −10 to *x* = −8, where *x* = −8 is not included in the domain. Therefore, you would express this interval as [−10, −8). Then find the next interval, which exists from *x* = −8 to *x* = −2; again, *x* = −8 is not included. Thus, this interval is expressed as (−8, −2]. Repeat the process for the next interval, which goes from *x* = −2 to *x* = −1, where −1 is not included. This interval is expressed as [−2, −1). Finally, find the last interval, which goes from *x* = −1 to *x* = 5, where *x* = −1 is not included. This interval is expressed as (−1, 5].

Now you need to join all of the intervals together with the union symbol to express the domain of *m*(*x*). Therefore, the domain of *m*(*x*) is [−10, −8) ∪ (−8, −2] ∪ [−2, −1) ∪ (−1, 5]. Since −2 is included in both the intervals (−8, −2] and [−2, −1), the domain can also be expressed as [−10, −8) ∪ (−8, −1) ∪ (−1, 5].

But what if the function goes on *forever?* Is that even possible? Of course it is! Consider the linear function *ƒ*(*x*) = *x* + 6:

Unlike the previous graphs, *g*(*x*) and *h*(*x*), the graph of *ƒ*(*x*) doesn’t have any endpoints. No worries—you can still define the domain and range of *ƒ*(*x*) using the infinity sign (∞) in your interval notation to represent intervals that extend indefinitely in one or both directions.

Since the function *ƒ*(*x*) = *x* + 6 extends infinitely in both directions, across all *x*- and *y*-values, both the domain and range of *ƒ*(*x*) would be represented by the interval (−∞, ∞).

Notice that the infinity

sign is not associated with

brackets. Since the infinity

symbol does not represent

a specific number, it will

always be accompanied

by a parenthesis.