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Describing Domain and Range

Sometimes it can be helpful to describe or analyze the set of all inputs and outputs for which a function is defined. These quantities are called domain and range, respectively.


The domain, is the set of all -values or inputs for which a function is defined. There are two reasons for numbers to be excluded from the domain:

  • The number gives a forbidden calculation, such as or
  • The function describes a specific situation. Suppose, for example, represents the price of apples. It does not make sense to consider the cost of apples. Thus, would not be in the domain of


The range, is the set of all -values or outputs a function gives. Since depends on the domain determines the range. Some functions can result in positive and negative -values, whereas others cannot. For example, consider For any input value, will show positive and negative outputs. The range of is all real numbers. Conversely, will only yield non-negative outputs, since the square of a number is never negative. Then, the range of is


The table describes the function,

Determine the domain and the range of the function.

Show Solution

The domain is the set of all -values for which the function is defined. We can find them in the left column. The range is the set of all -values, and we find them in the right column. Thus, the range of the function is


Use the graph to determine the domain and the range of the function.

Show Solution

Since it's impossible to draw an infinitely large coordinate system, we cannot sketch the entire graph. However, it's reasonable to assume it continues in the same manner beyond the drawn region. In this case, the graph will continue infinitely to the right and infinitely upward. Thus, the domain and the range do not have an upper limit. However, we can determine their lower limits.

The graph of the function begins at so the domain includes all numbers greater than or equal to This is written as

Similarly, we see that the smallest -value is so the range includes all numbers greater than or equal to This is written Thus, the domain and range of are as shown.


A theater has a square stage, and each side of the stage floor is meters. A circular rug is to be laid out on the stage floor. Create a function that describes the area of the rug, and determine its domain and range.

Show Solution

Let's start by making a rough sketch of the situation. We're confined to the stage's measurements because the rug cannot be bigger than the stage. We'll name the radius of the rug

The area of a circle is given by the formula where is the radius of the circle. Since radius measures the distance from the circle to its center, the radius of the rug must be greater than Additionally, since the length across the entire circle — the diameter — must not be greater than meters, the maximum value of is meters. This gives the domain To find the range, we determine the minimum and the maximum value of the area using the domain above. If the radius is meters the area is also To find the maximum value of the area, we'll substitute for
The area of the rug can range between and approximately square meters. Thus, the range is
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