**Domain and Range**

Before beginning a discussion of domain and range, one must understand the idea of a function. This topic is covered more extensively in a subsequent tutorial entitled Functions, but for our purposes, we will define a function to be an expression which takes an input value and generates a *single* output value. For example, the following are functions because in each case, a choice of input value *x* leads to a single output value *y*.

Two concepts that are of great importance in the study of functions are *domain* and *range*.

__Domain__

The collection of all values of the input variable for which the function is defined.

__Range__

The collection of all output values that are obtained from using the domain values as inputs.

__Example 1__

Determine the domain and range of the following function:

__Solution 1__

Let’s start with the domain. We ask the question, “Are there any values of *x* for which the function is undefined?” In this case, the answer is no; no matter what value of *x* is chosen, the function always yields a well defined value for *y*. Therefore, we say that the domain of this function is the set of all real numbers.

In determining the range, we ask ourselves the question, “Are there any output values, *y,* that can never be generated no matter what we choose for the value of *x?*” A close examination of the function tells us that there are. Note that the *x* is squared and the square of a real number is always positive (or zero). So if the lowest that gets is zero, then the lowest that *y* can get is 1. On the other hand, there is no limit to how big *y* can get since we have placed no restrictions on the size of *x*. Therefore, we say that the range of this function is all real numbers greater than or equal to 1 or we write:

Range: {*y* | *y* ≥ 1}.

Note that it is also easy to see this range by looking at the graph of . Clearly the lowest point on the graph is the point (0,1) and the rest of the graph goes upward endlessly. This information shows that the smallest *y*-value is 1 and that the range is the set of all real numbers greater than or equal to 1.

__Example 2__

Determine the domain and range of the following function:

__Solution 2__

We begin again by looking for *x*-values which make this function undefined. In this case, there is one such value. Since division by zero is not allowed in mathematics, *x* cannot equal 2. Therefore, we say that the domain of this function is all real numbers except 2 or we write:

Domain: { *x* | *x* ≠ 2 }.

Then we determine if there are any *y*-values which can never be achieved as output values. A graph of the function is helpful at this point (see tutorial entitled Common Curves), but in the absence of a graph, we ask questions about the behavior near 2 and as we tend to positive and negative infinity. Try plugging in numbers like 2.1 and 2.001 for *x*. Notice how as we select *x*-values closer and closer to 2, the value of *y* gets larger and larger. If we use *x*-values very close to 2 but slightly less than 2, such as 1.9 and 1.999, the *y*-values get very large negatively. We can also see that for *x*-values that are very large, such as 10,000 or 1,000,000,000, the value of *y* becomes very, very small – near 0. The same holds true for negatively large *x*-values, such as -10,000 or
-1,000,000,000. So we have established that y can be very large -- positively or negatively -- and we have established that y can get very close to 0. But can y ever exactly equal 0? In order for a fraction to equal zero, the numerator must equal 0. Since this fraction’s numerator is always 1 and never zero, then y can never be zero. However, the only way for a fraction to equal zero is for the numerator to be zero. Therefore, we say that the range of this function is all real numbers except zero or we write:

Range: { *y* | *y* ≠ 0 }.