In this explainer, we will learn how to identify the domain and range of functions from their equations.
Recall that a function assigns to members of a one set, say , values in a second set, . We think of as the source of inputs to the function and as the target of its outputs. So, for an input from , the output given by is the value .
Definitions: Domain and Range
- The domain of is the set of inputs for which is defined.
- The range of is the set of all outputs as wanders over the domain.
Notice that the range is a subset of .
Generally, the domain is all of , particularly when the function is given as a set of ordered pairs or by a diagram.
For example, the following defines a function into the set :
- The domain of is the set .
- The range of is the set of all values taken by , and is therefore a proper subset of .
The instances where the domain needs to be identified are usually when the function is defined for subsets of the real numbers. So, you will meet questions like, “What is the domain of ?”
What this question is asking is, “What is the largest subset of the real numbers for which the assignment makes sense?” The solution, of course, is all the nonzero numbers, the set . Or, put another way, the set .
Sometimes, domains for functions between sets of real numbers must be determined from a graph, such as the following example.
Example 1: Determining the Domain of a Function from Its Graph
Determine the domain of the function represented in the shown graph.
The solid dot at the extreme left of the curve tells us that , and the arrow at the other end says that the function is defined for real numbers to the right of the axis. Therefore, the domain of this function is
More often, determining the domain of a function defined by a formula involves deciding what the permitted inputs are.
Example 2: Determining the Domain of a Radical Function
Determine the domain of the function .
The square root is only valid for nonnegative numbers, so we require or
If , this means (i.e., ).
If , then, by definition, , which is the same as .
So, our domain is the set
Of course, this is not an interval, although you can write it as the complement of one: .
Here is another example.
Example 3: Finding the Domain of a Piecewise Defined Function
Determine the domain of the function
In this case, the figure is not even necessary to answer the question. lies in the domain, provided that it satisfies one of the conditions of the piecewise definition. This means that belongs to the union which we see is actually the same as .
Suppose that is a function on a domain . By definition, the range of is the set of all its outputs: . If is part of the real line and is represented by a graph, this range is visible as part of the -axis.
Example 4: Finding the Range of a Quadratic Function from Its Graph
Determine the range of the function represented by the shown graph.
We can see that if this is the function , then , and this is the maximum value of the function. There is no with . At the same time, the graph appears to be a parabola that takes on all values less than or equal to 1. The range is
Even when we are not supplied with a graph, if we have a formula describing a function, we can determine the range from that. For example, with above, we saw that the numbers were important for the domain. So, . Similarly, .
We guess that the range of is just all nonnegative numbers, the interval . This is the claim that any for some . For example, suppose we took , what is so that
In other words, what makes
The answer is found by solving the equation. We need . So, , and we have two values with target 1,234: and . It is easy to see why any will do.
Example 5: Finding the Range of a Linear Function from Its Domain
If , where , find the range of .
The graph of is a straight line, so we know that for , the values all lie between the values at the endpoints, which are included. These values are
In other words,