While the formal definition of a function is a set of ordered pairs, there are many ways that we represent functions when studying them. Each representation has advantages and disadvantages and being able to change between different representations of the same function is an important skill. Let’s look at some different types of representations.
Functions on a finite set of points are often represented by tables. One advantage of a table is that you can easily see all the information for a function. One disadvantage is if you have too many input values, it can be difficult to analyze them all in a table format. The exploration below shows and example of a function given by a table.
If the set of ordered pairs that makes up your function is an infinite set, you cannot represent your entire function as a table because it would go on forever! Instead, you might see a table with a sampling of points from a function. In that case, unless you have additional information about your function, you cannot know what the outputs are for inputs not listed in the table. In fact, without additional information, you cannot even say which values are allowed to be inputs for the function!
Arrow diagrams are another tool used to represent functions. Here is an arrow diagram that corresponds to the function in the exploration above.
Arrow diagrams can help make it easier to see when multiple inputs go to the same output. Arrow diagrams can also be used to represent relations, and can make it easier to see if the relation is a function. The three arrow diagrams below show relations. The first two relations are functions, but the last relation is not a function because the same input goes to two different outputs.
Graphing points on a coordinate plane is a great way to represent a function and one which is used often. Graphs are most often used for functions where both the inputs and outputs are numbers. When graphing functions, typically the horizontal axis represents the input values and the vertical axis represents the output values.
The line in the graph above is the set of all points of the form for any real number value of .
Let’s use this test to analyze each of the graphs in this example.
Therefore, by the vertical line test, this graph is not the graph of a function. Really, the vertical line we found at is helping us to quickly identify that both and are points on this graph so that for the same input, , there are two different outputs, and .
Note that we cannot draw all possible vertical lines. Really, we only need to look at the points on this graph which are not a line. We already said the graphs of lines (without any other points or curves above or below that line) are functions. So on this graph, the interesting points we want to look at more closely are , , and . In each of these cases, there is either one or no outputs. In particular, for the corresponding output is . There is an open dot at so this point is not in the relation. Similarly, is a point on this graph but is not. At , there are no corresponds outputs. We say are function is not defined at .
You could consider this a graph of five separate functions all graphed on the same coordinate axes, but that is a different question from the one being asked.
Another common way to represent functions is using a formula. In the example of Dolbear’s Law, the function which models this law is given as a formula by When we give functions as a formula, we also need to say which input values are allowed, that is, what the domain of the function is. In this case, the domain is . If the domain is not given, the domain is assumed to be all the values for which the formula used to defined the function makes sense, that is, the implied domain.
Here are some examples of functions represented by a formula.
Recall that this is the famous function named sine.
Here is an example of working with the formula representation of a function.
Similarly, means to evaluate the function at the input . That is, . This expression contains no like terms, so it cannot be simplified further.
. Distributing, we find that , so .
, and multiplying yields , so .
Although involves an unusual symbol, we can treat the symbol like a number. Thus, .
Zero is a very important number, and as we will see later, knowing where a function’s - or -value equals zero can be powerful information.
An -intercept of is a point such that . That is, a point in which the graph of the function touches the -axis.
The -intercept of is a point such that . That is, a point in which the graph of the function touches the -axis. Unlike -intercepts, there can be only one -intercept for each function.