3.2. Domain and Range
Domain and Range
One of our main goals in mathematics is to model the real world with mathematical functions. In doing so, it is important to keep in mind the limitations of those models we create.
This table shows a relationship between circumference and height of a tree as it grows.
| Circumference, c | 1.7 | 2.5 | 5.5 | 8.2 | 13.7 |
| Height, h | 24.5 | 31 | 45.2 | 54.6 | 92.1 |
While there is a strong relationship between the two, it would certainly be ridiculous to talk about a tree with a circumference of -3 feet, or a height of 3000 feet. When we identify limitations on the inputs and outputs of a function, we are determining the domain and range of the function.
Example 3.2.1
Using the tree table above, determine a reasonable domain and range.
We could combine the data provided with our own experiences and reason to approximate the domain and range of the function h = f(c). For the domain, possible values for the input circumference c, it doesn’t make sense to have negative values, so c > 0. We could make an educated guess at a maximum reasonable value, or look up that the maximum circumference measured is about 119 feet. With this information we would say a reasonable domain is 0 < c ≤ 119 feet.
Similarly for the range, it doesn’t make sense to have negative heights, and the maximum height of a tree could be looked up to be 379 feet, so a reasonable range is 0 < h ≤ 379 feet.
Example 3.2.2
When sending a letter through the United States Postal Service, the price depends upon the weight of the letter, as shown in the table below. Determine the domain and range.
| Letters | |
| Weight not Over | Price |
| 1 ounce | $0.44 |
| 2 ounces | $0.61 |
| 3 ounces | $0.78 |
| 3.5 ounces | $0.95 |
Suppose we notate Weight by w and Price by p, and set up a function named P, where Price, p is a function of Weight, w. p = P(w).
Since acceptable weights are 3.5 ounces or less, and negative weights don’t make sense, the domain would be 0 < w ≤ 3.5. Technically 0 could be included in the domain, but logically it would mean we are mailing nothing, so it doesn’t hurt to leave it out.
Since possible prices are from a limited set of values, we can only define the range of this function by listing the possible values. The range is p = $0.44, $0.61, $0.78, or $0.95.
Notation
In the previous examples, we used inequalities to describe the domain and range of the functions. This is one way to describe intervals of input and output values, but is not the only way.
Using inequalities, such as 0 < c ≤ 163 , 0 < w ≤ 3.5 , and 0 < h ≤ 379 imply that we are interested in all values between the low and high values, including the high values in these examples.
However, occasionally we are interested in a specific list of numbers like the range for the price to send letters, p = $0.44, $0.61, $0.78, or $0.95. These numbers represent a set of specific values: {0.44, 0.61, 0.78, 0.95}
Representing values as a set, or giving instructions on how a set is b