8.6 Domain and Range of a Function
Learning Objectives
- I can build on your previous knowledge of functions by learning how to create a table of values for a function.
- I can identify the domain and range of a function.
Introduction
Suppose you have a function that allows you to input the number of years you have until retirement and which outputs the amount of money you should have saved. How would you go about determining the domain of such a function? How would you decide on the range? After completing this concept, you'll be able to create a table of values for a function like this and give its domain and range.
Guided Learning
Using a Function to Generate a Table
A function really is an equation. Therefore, a table of values can be created by choosing values to represent the independent variable. The answers to each substitution represent .
Example A
Use Joseph’s function to generate a table of values. Because the variable represents the number of rides Joseph will pay for, negative values do not make sense and are not included in the list of values of the independent variable.
Solution:
R | J(r) = 2r |
0 | 2(0) = 0 |
1 | 2(1) = 2 |
2 | 2(2) = 4 |
3 | 2(3) = 6 |
4 | 2(4) = 8 |
5 | 2(5) = 10 |
6 | 2(6) = 12 |
As you can see, the list cannot include every possibility. A table allows for precise organization of data. It also provides an easy reference for looking up data and offers a set of coordinate points that can be plotted to create a graphical representation of the function. A table does have limitations; namely it cannot represent infinite amounts of data and it does not always show the possibility of fractional values for the independent variable.
Domain and Range of a Function
The set of all possible input values for the independent variable is called the domain. The domain can be expressed in words, as a set, or as an inequality. The values resulting from the substitution of the domain represent the range of a function.
The domain of the function representing Joseph’s situation will not include negative numbers because it does not make sense to ride negative rides. He also cannot ride a fraction of a ride, so decimals and fractional values do not make sense as input values. Therefore, the values of the independent variable r will be whole numbers beginning at zero.
Domain: All whole numbers
The values resulting from the substitution of whole numbers are whole numbers times two. Therefore, the range of the function representing Joseph’s situation is still whole numbers, just twice as large.
Range: All even whole numbers
Example B
A tennis ball is bounced from a height and bounces back to 75% of its previous height. Write the function for this scenario and determine its domain and range.
Solution: The function representing this situation is , where represents the previous bounce height.
Domain: The previous bounce height can be any positive number, so .
Range: The new height is 75% of the previous height, and therefore will also be any positive number (decimal or whole number), so the range is all positive real numbers.
Example C
Find the range of when the domain is .
Solution:
Since the range is the output, we plug in the values in the domain to see what values we will get out.
The range for the given domain is .
Notice that we used function notation to keep track of which input value gave us which output value. This will be useful later.
Guided Practice
Eli makes $20 an hour tutoring math.
a. Write a function expressing the amount of money she earns.
b. What are the domain and range of this function?
c. Suppose Eli will only work for either 1, 1.5, or 2 hours. Express this domain and the corresponding range in a table.
Check your work with a partner.
Solutions:
a. Let represent money earned for hours. Then the function is .
b. Since hours worked can only be zero or positive, is the domain. If Eli works for zero hours, she will earn zero dollars. She could also earn any positive amount of money, so the range is also all non-negative real numbers. That is, .
c. First we plug the domain into our function:
Putting this into a table, we get:
h | M(h) |
1 | 20 |
1.5 | 30 |
2 | 40 |
Review
Domain
The domain of a function is the set of all possible input values for that function.
Range
The range of the function is the set of all possible output values for that function.