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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 }.