7.15: Domain and Range of a Function
Remember Kara and Marc? Well, they are still in Boston with their grandparents.
Kara and Marc decided to spend one morning helping out at a car wash to benefit a club that their grandparents have participated in for year.
The car wash was a busy place. At the beginning there weren’t any cars, but between 9 am and 10 am the class washed 5 cars. From 10 to 11, the class washed 10 cars, from 11 to 12 the class washed 15 cars and from 12 – 1 the class washed 20 cars.
Marc kept track of the cars washed each hour.
Hour 1 - 5 cars
Hour 2 - 10 cars
Hour 3 - 15 cars
Hour 4 - 20 cars
Can you write this list as ordered pairs? Can you identify the domain and range of the function?
This Concept will teach you how to do this.
Guidance
We use the word “function” all the time in everyday speech. We say things like “It’s a function of time” or “It’s a function of price.” This is a real life application of a mathematical concept called a function. You will learn how to apply functions to real-world examples, but first let’s look at what a function is and how we can understand it better.
What is a function?
A function is a set of ordered pairs in which the first element in any pair corresponds to exactly one second element.
For example, look at this set of ordered pairs. Notice that braces, {}, are used to surround the set of ordered pairs.
\begin{align*}& \{ (0, 5), (\underline{1}, \underline{6}), (2, 7), (3, 8)\}\\ & \quad \ \ \qquad \uparrow \uparrow\end{align*}
In (1, 6), 1 is the first element and
6 is the second element.
Each of the first elements––0, 1, 2, and 3––corresponds to exactly one second element. So, this set of ordered pairs represents a function.
Let's take a look at another set of ordered pairs.
\begin{align*}& \{ (\underline{2}, 4), (5, 3), (6, 7), (\underline{2}, 8)\}\\ & \ \ \Box \qquad \qquad \qquad \quad \ \Box\end{align*}
The first element, 2, corresponds
to two different second elements––4 and 8.
Since one of the first elements corresponds to two different second elements, the set of ordered pairs above does not represent a function.
We can use these criteria to determine whether or not a series of ordered pairs forms a function.
Now that you know how to identify a function, let’s look at some of the key words associated with functions.
- Domain
- Range
The domain of a function is the set of all the first elements in a function. The range is the set of all the second elements in a function.
Let’s look at a set of ordered pairs and identify the domain and range of the series.
The ordered pairs below represent a function
{(0, -10), (2, -8), (4, -6), (6, -4)}
a. Identify the domain of the function.
b. Identify the range of the function.
Consider part \begin{align*}a\end{align*} first.
The domain is the set of all the first elements in the function. These first elements are underlined below.
\begin{align*}\{ (\underline{0}, -10), (\underline{2}, -8), (\underline{4}, -6), (\underline{6}, -4)\}\end{align*}
The domain of this function is {0, 2, 4, 6}.
Next, consider part \begin{align*}b\end{align*}.
The range is the set of all the second elements in the function. These second elements are underlined below.
\begin{align*}\{ (0, \underline{-10}), (2, \underline{-8}), (4, \underline{-6}), (6, \underline{-4})\}\end{align*}
The range of this function is {-10, -8, -6, -4}.
Now it's your turn. Identify the domain and range of each function.
Example A
(1, 3) (2, 4) (5, 7) (9, 11)
Solution: Domain \begin{align*}\{1, 2, 5, 9\}\end{align*}, Range \begin{align*}\{3, 4, 7, 11\}\end{align*}
Example B
(8, 12) (9, 22) (4, 7) (2, 5)
Solution: Domain \begin{align*}\{8, 9, 4, 2\}\end{align*}, Range \begin{align*}\{12, 22, 7, 5\}\end{align*}
Example C
(8, 9) (3, 5) (7, 6) (10, 12)
Solution: Domain \begin{align*}\{8, 3, 7, 10\}\end{align*}, Range \begin{align*}\{9, 5, 6, 12\}\end{align*}
Here is the original problem once again.
Kara and Marc decided to spend one morning helping out at a car wash to benefit a club that their grandparents have participated in for year.
The car wash was a busy place. At the beginning there weren’t any cars, but between 9 am and 10 am the class washed 5 cars. From 10 to 11, the class washed 10 cars, from 11 to 12 the class washed 15 cars and from 12 – 1 the class washed 20 cars.
Marc kept track of the cars washed each hour.
Hour 1 - 5 cars
Hour 2 - 10 cars
Hour 3 - 15 cars
Hour 4 - 20 cars
Can you write this list as ordered pairs? Can you identify the domain and range of the function?
First, use the hours as the domain and the number of cars as the range.
(1, 5) (2, 10) (3, 15) (4, 20)
The domain is {1, 2, 3, 4}.
The range is {5, 10, 15, 20}.
This is our answer.
Vocabulary
Here are the vocabulary words in this Concept.
- Function
- A set of ordered pairs in which one element corresponds to exactly one other element. Functions can be expressed as a set of ordered pairs or in a table.
- Domain
- the \begin{align*}x\end{align*} value of an ordered pair or the \begin{align*}x\end{align*} values in a set of ordered pairs.
- Range
- the \begin{align*}y\end{align*} value of an ordered pair or the \begin{align*}y\end{align*} values in a set of ordered pairs.
Guided Practice
Here is one for you to try on your own.
Write the domain and range of this set.
(1, 3) (3, 9) (4, 6) (5, 12)
Answer
The first value of each ordered pair represents the domain.
Domain {1, 3, 4, 5}
The second value of each ordered pair represents the range.
Range {3, 9, 6, 12}
This is our answer.
Video Review
Here is a video for review.
- This is a Khan Academy video on the domain and range of a function.
Practice
Directions: Identify whether or not each series of ordered pairs forms a function.
1. (1, 3)(2, 6)(2, 5) (3, 7)
2. (2, 5) (3, 6) (4, 7) (5, 8)
3. (6, 1) (7, 2) (8, 3)
4. (5, 2) (5, 3) (5, 4) (5, 5)
5. (81, 19)(75, 18) (76, 18) (77, 19)
Directions: Name the domain in numbers 1 – 5.
6.
7.
8.
9.
10.
Directions: Name the range in numbers 1 – 5.
11.
12.
13.
14.
15.
domain
The domain of a function is the set of -values for which the function is defined.Range
The range of a function is the set of values for which the function is defined.Continuous
Continuity for a point exists when the left and right sided limits match the function evaluated at that point. For a function to be continuous, the function must be continuous at every single point in an unbroken domain.dependent variable
The dependent variable is the output variable in an equation or function, commonly represented by or .Discrete
A relation is said to be discrete if there are a finite number of data points on its graph. Graphs of discrete relations appear as dots.Formula
A formula is a type of equation that shows the relationship between different variables.Function
A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for .independent variable
The independent variable is the input variable in an equation or function, commonly represented by .Integer
The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...Real Number
A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.Image Attributions
Here you'll learn to identify the domain and range of a simple linear function.
Concept Nodes:
domain
The domain of a function is the set of -values for which the function is defined.Range
The range of a function is the set of values for which the function is defined.Continuous
Continuity for a point exists when the left and right sided limits match the function evaluated at that point. For a function to be continuous, the function must be continuous at every single point in an unbroken domain.dependent variable
The dependent variable is the output variable in an equation or function, commonly represented by or .Discrete
A relation is said to be discrete if there are a finite number of data points on its graph. Graphs of discrete relations appear as dots.Formula
A formula is a type of equation that shows the relationship between different variables.Function
A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for .independent variable
The independent variable is the input variable in an equation or function, commonly represented by .Integer
The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...Real Number
A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.