2.9: Finding the Domain and Range of Functions
You just got a new parttime job at the mall that pays a base rate of $150/week plus $5/sale. Your boss encourages you to make as many sales as possible but she will cap your weekly earnings at $250. What are the domain and range of the function represented by this situation?
Watch This
James Sousa: Ex 2: Determine the Domain and Range of the Graph of a Function
Guidance
The input and output of a function is also called the domain and range. The domain of a function is the set of all input values. The range of a function is the set of all output values. Sometimes, a function is a set of points. In this case, the domain is all the
Example A
Determine if {(9, 2), (7, 3), (4, 6), (10, 4), (2, 7)} is a function. If so, find the domain and range.
Solution: First, this is a function because the
The
Example B
Find the domain and range of
Solution: Because this is a linear equation we also know that it is a linear function, from the previous concept. All lines continue forever in both directions, as indicated by the arrows.
Notice the line is solid, there are no dashes or breaks, this means that it is continuous. A continuous function has a value for every
Domain:
In words,
The second option,
The range of this function is also continuous. Therefore, the range is also the set of all real numbers. We can write the range in the same ways we wrote the domain, but with
Range:
Example C
Find the domain and range of the graphed function below.
Solution: This is a function, even though it might not look like it. This type of function is called a piecewise function because it pieces together two or more parts of other functions.
To find the domain, look at the possible
Mathematically, this would be written:
However, upon further investigation, the branch on the left does pass through the yellow region, where we though the function was not defined. This means that the function is defined between 1 and 3 and thus for all real numbers. However, below 3, there are no
Intro Problem Revisit The function represented by this situation can be written as
Therefore, the domain of the function is
To find the range, plug the two extremes of the domain into the equation. When x equals 0, y equals 150, and when x equals 20, y equals 250.
Therefore the range of the function is
Vocabulary
 Domain
 The input of a function.
 Range
 The output of a function.
 Continuous Function
 A function without breaks or gaps.
 Interval Notation

The notation
[a,b) , where a function is defined betweena andb . Use ( or ) to indicate that the end value is not included and [ or ] to indicate that the end value is included. Infinity and negative infinity are never included in interval notation.
 Piecewise Function
 A function that pieces together two or more parts of other functions to create a new function.
\begin{align*}\infty\end{align*} and \begin{align*}\infty:\end{align*} The symbols for infinity and negative infinity, respectively.
Guided Practice
Find the domain and range of the following functions.
1. {(8, 3), (4, 2), (6, 1), (5, 7)}
2. \begin{align*}y = \frac{1}{2}x + 4\end{align*}
3.
4.
Answers
1. Domain: \begin{align*}x \in \{8,4,6,5\}\end{align*} Range: \begin{align*}y \in \{3,2,1,7\}\end{align*}
2. Domain: \begin{align*}x \in \mathbb{R}\end{align*} Range: \begin{align*}y \in \mathbb{R}\end{align*}
3. This is a piecewise function. The \begin{align*}x\end{align*}values are not defined from 2 to 1. The range looks like it is not defined from 1 to 7, but the lines continue on, filling in that space as \begin{align*}x\end{align*} gets larger, both negatively and positively.
Domain: \begin{align*}x \in (\infty,2)\cup (1,\infty)\end{align*} Range: \begin{align*}y \in \mathbb{R}\end{align*}
4. This is a parabola, the graph of a quadratic equation. Even though it might not look like it, the ends the graph continue up, infinitely, and \begin{align*}x\end{align*} keeps growing. In other words, \begin{align*}x\end{align*} is not limited to be between 9 and 5. It is all real numbers. The range, however, seems to start at 6 and is all real numbers above that value.
Domain: \begin{align*}x \in \mathbb{R}\end{align*} Range: \begin{align*}y \in [6,\infty)\end{align*}
Practice
Determine if the following sets of points are functions. If so, state the domain and range.
 {(5, 6), (1, 5), (7, 3), (0, 9)}
 {(9, 8), (7, 8), (7, 9), (8, 8)}
 {(6, 2), (5, 6), (5, 2)}
 {(1, 2), (6, 3), (10, 7), (8, 11)}
 {(5, 7), (3, 7), (5, 8), (8, 1)}
 {(3, 4), (5, 6), (1, 2), (2, 6)}
Find the domain and range of the following functions.
 \begin{align*}y = 3x  7\end{align*}
 \begin{align*}6x 2y = 10\end{align*}
 Challenge
 Writing Make a general statement about the domain and range of all linear functions. Use the proper notation.
Notes/Highlights Having trouble? Report an issue.
Color  Highlighted Text  Notes  

Please Sign In to create your own Highlights / Notes  
Show More 
closed interval
A closed interval includes the minimum and maximum values (endpoints) of the interval.Continuous Function
A continuous function is a function without breaks or gaps. It contains an infinite, uncountable number of values.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 .open interval
An open interval does not include the endpoints of the interval.Piecewise Function
A piecewise function is a function that pieces together two or more parts of other functions to create a new function.Relation
A relation is any set of ordered pairs . A relation can have more than one output for a given input.union
is a symbol that stands for union and is used to connect two groups together. It is associated with the logical term OR.Image Attributions
Here you'll learn how to find the domain and range of certain functions and sets of points.