1.10: Domain and Range of a Function
What if you were given a rule that relates two variables, like
Watch This
CK12 Foundation: 0110S Introduction to Functions
For another look at the domain of a function, see the following video, where the narrator solves a sample problem from the California Standards Test about finding the domain of an unusual function.
Khan Academy: CA Algebra I: Functions
Guidance
A function is a rule for relating two or more variables. For example, the price you pay for phone service may depend on the number of minutes you talk on the phone. We would say that the cost of phone service is a function of the number of minutes you talk. Consider the following situation.
Josh goes to an amusement park where he pays $2 per ride.
There is a relationship between the number of rides Josh goes on and the total amount he spends that day: To figure out the amount he spends, we multiply the number of rides by two. This rule is an example of a function. Functions usually—but not always—are rules based on mathematical operations. You can think of a function as a box or a machine that contains a mathematical operation.
Whatever number we feed into the function box is changed by the given operation, and a new number comes out the other side of the box. When we input different values for the number of rides Josh goes on, we get different values for the amount of money he spends.
The input is called the independent variable because its value can be any number. The output is called the dependent variable because its value depends on the input value.
Functions usually contain more than one mathematical operation. Here is a situation that is slightly more complicated than the example above.
Jason goes to an amusement park where he pays $8 admission and $2 per ride.
The following function represents the total amount Jason pays. The rule for this function is "multiply the number of rides by 2 and add 8."
When we input different values for the number of rides, we arrive at different outputs (costs).
These flow diagrams are useful in visualizing what a function is. However, they are cumbersome to use in practice. In algebra, we use the following shorthand notation instead:
First, we define the variables:
So,
In algebra, the notations
Identify the Domain and Range of a Function
In the last example, we saw that we can input the number of rides into the function to give us the total cost for going to the amusement park. The set of all values that we can use for the input is called the domain of the function, and the set of all values that the output could turn out to be is called the range of the function. In many situations the domain and range of a function are both simply the set of all real numbers, but this isn’t always the case. Let's look at our amusement park example.
Example A
Find the domain and range of the function that describes the situation:
Jason goes to an amusement park where he pays $8 admission and $2 per ride.
Solution
Here is the function that describes this situation:
In this function,
 The values have to be zero or positive, because Jason can't go on a negative number of rides.
 The values have to be integers because, for example, Jason could not go on 2.25 rides.
 Realistically, there must be a maximum number of rides that Jason can go on because the park closes, he runs out of money, etc. However, since we aren’t given any information about what that maximum might be, we must consider that all nonnegative integers are possible values regardless of how big they are.
Answer For this function, the domain is the set of all nonnegative integers.
To find the range of the function we must determine what the values of
Answer The range of this function is the set of all even integers greater than or equal to 8.
Example B
Find the domain and range of the following functions.
a) A ball is dropped from a height and it bounces up to 75% of its original height.
b)
Solution
a) Let’s define the variables:
A function that describes the situation is
Answer
The domain is the set of all real numbers greater than zero. The range is also the set of all real numbers greater than zero.
b) Since there is no word problem attached to this equation, we can assume that we can use any real number as a value of
Answer
The domain of this function is all real numbers. The range of this function is all nonnegative real numbers.
In the functions we’ve looked at so far,
These expressions all represent the same function: a function where the dependent variable is three times the independent variable. Only the symbols are different. In practice, we usually pick symbols for the dependent and independent variables based on what they represent in the real world—like
Make a Table For a Function
A table is a very useful way of arranging the data represented by a function. We can match the input and output values and arrange them as a table. For example, the values from Example 1 above can be arranged in a table as follows:
A table lets us organize our data in a compact manner. It also provides an easy reference for looking up data, and it gives us a set of coordinate points that we can plot to create a graph of the function.
Example C
Make a table of values for the function
Solution
Make a table of values by filling the first row with the input values and the next row with the output values calculated using the given function.
When you’re given a function, you won’t usually be told what input values to use; you’ll need to decide for yourself what values to pick based on what kind of function you’re dealing with. We will discuss how to pick input values throughout these lessons.
Watch this video for help with the Examples above.
CK12 Foundation: Introduction to Functions
Vocabulary
A function is a rule for relating two or more variables, one of which is the input variable and the other is the output variable. The input is called the independent variable because its value can be any number. The output is called the dependent variable because its value depends on the input value. The set of all values that we can use for the input is called the domain of the function, and the set of all values that the output could turn out to be is called the range of the function.
Guided Practice
Identify the domain and then make a table of values for the function
Solution
Since you cannot compute the square root of negative numbers, these cannot be in the domain. Since we cannot have 0 in the denominator, 0 is also not in the domain. This means that the domain is all real numbers greater than zero.
Make a table of values by filling the first row with the input values and the next row with the output values calculated using the given function.
Practice
For 16, identify the domain and range of the following functions.
 Dustin charges $10 per hour for mowing lawns.
 Maria charges $25 per hour for tutoring math, with a minimum charge of $15.

f(x)=15x−12 
f(x)=2x2+5 
f(x)=1x 
f(x)=x√3  What is the range of the function
y=x2−5 when the domain is 2, 1, 0, 1, 2?  What is the range of the function
y=2x−34 when the domain is 2.5, 1.5, 5?  What is the domain of the function
y=3x when the range is 9, 12, 15?  What is the range of the function
y=3x when the domain is 9, 12, 15?  Angie makes $6.50 per hour working as a cashier at the grocery store. Make a table that shows how much she earns if she works 5, 10, 15, 20, 25, or 30 hours.
 The area of a triangle is given by the formula
A=12bh . If the base of the triangle measures 8 centimeters, make a table that shows the area of the triangle for heights 1, 2, 3, 4, 5, and 6 centimeters.  Make a table of values for the function
f(x)=2√x+3 for input values 1, 0, 1, 2, 3, 4, 5.
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.domain
The domain of a function is the set of values for which the function is defined.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...Range
The range of a function is the set of values for which the function is defined.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
Description
Learning Objectives
Here you'll learn how to find the domain and range of a function and you'll make a table of values for a given function.