Different Representations of a Function

Definition of the Range of a Function
As we said, functions have inputs and outputs. The inputs are what we put in the function, and the outputs are what come out. We call the set of outputs of a function the range of a function. For example, consider the function defined by the rule that we take an input and raise it to the third power. This can be represented in equation form as y = x^3, and when this function is given input values of {2, 1, 0, 1, 2}, we can find the corresponding outputs by plugging those inputs in for 'x' in the equation.
For instance, if we input 2, we have y = (2)^3 = 8, so when the input is 2, the output is 8, and 8 is in our range. When we find each of the corresponding outputs to our inputs, we have our range. In our example, the range is {8, 1, 0, 1, 8}, because these are the outputs corresponding to 2, 1, 0, 1, and 2 respectively.
Finding Range of a Function
To find the range of a function, we simply need to find the functions outputs, based on its inputs. As we mentioned, functions can be represented in different ways. Let's look at some different examples of finding the range of a function for different representations of a function.
1.First, let's consider our initial drink menu example. We described this function using words. The function's rule assigns a small drink to $1.50, a medium drink to $2.50, and a large drink to $3.50. The inputs are the drink size, and the outputs are the drink price. The range of this function is the set of all the outputs. Therefore, the range for this function is the set of all outputs, or {$1.50, $2.50, $3.50}.
2.Consider the following function represented in the table.
Input 
Output 
3 
6 
2 
4 
1 
2 
0 
0 
1 
2 
2 
4 
3 
6 
By definition the range is the set of all the outputs of a function, so to find the range, we simply list the outputs {6, 4, 2, 0, 2, 4, 6}.
3.Observe this graph:
Example

The function represented by this graph has its xvalues as its inputs, and its yvalues as its outputs. That is, given an xvalue, we can determine the corresponding yvalue by looking at the graph. Therefore, the range of this function consists of all the yvalues where the function is defined (this is where the graph has a yvalue, or the yvalues where there is a picture). We can see that the graph takes on yvalues from 1 to 4. Therefore, the range of this function consists of all the numbers from 1 to 4, including 1 and 4.
4.Lastly, let's look at the function represented by the equation y = x^2. The inputs of this function consist of all real numbers. This function takes an input and squares it to give the output. Notice that when we square a number, we always end up with a positive number. Therefore, our range will consist of all positive real numbers. It will also contain the number 0, because if we input 0, we get 0 as an output. This can also be observed by looking at the graph of y = x^2.
Graph of y = x^2

Notice the graph has a picture for all y values greater than or equal to zero. Therefore the range consists of all real numbers greater than or equal to zero.
Lesson Summary
The range of a function is the set of all outputs of that function. To find the range of a function, we simply find the outputs of the function. It is useful to know the range of a function, because this allows us to know what values will come out of the function and what to expect. This lets us predict how certain phenomena represented by functions will behave. As we can see, knowing what the range of a function is and how to find that range is extremely useful.