There are many algebra books that have many ways of defining a function. Chances are if you look at four different books you would find at least two different explanations of a function. Does this mean
some explanations are right and others are wrong or that there are numerous definitions for a function? Not really. The definition
of a function
never changes, but the way teachers and textbooks explain
that definition take on many forms.
We’re going to look at functions in the following way.
A function is a relationship that meets certain conditions between two variables. These variables are usually called the independent variable and the dependent variable. In a function, each value for the independent variable corresponds to exactly one value for the dependent variable.
If a relationship with independent and dependent variables but not necessarily with one dependent variable for each independent variable, that relationship is called a relation. Notice that a function and a relation both describe relationships between two variables, but a function is just a special form of a relation.
One of the easiest ways to look at functions and relations is by looking at ordered pairs of numbers
Consider the following set of ordered pairs:
When dealing with ordered pairs, the independent variable is listed first and the dependent variable is listed second. Notice that each independent variable (0, 1, 2, and 3) is only paired up with exactly one dependent variable. This set of ordered pairs would be considered a function.
When we noted that the dependent variables took on values of 0, 1, 2, and 3 we created a list of values that we normally refer to as the domain of a function. Although we did not list the dependent variables, we could do that by listing 1, -2, 0, and 2. We normally refer to these values as the range of the function.
Using our new terminology we have:
domain of So now we have another way to look at our explanation of a function. See how easy it is to come up with so many varied ways to describe the same thing?
We can now say that each value in the domain of a function can only be paired with one value from the range of the function.
Now look at a different set of ordered pairs.
We can find the domain of . Notice that we do not have to list the value of 1 more than once, but because it appears more than once in the list of ordered pairs, we should be suspicious that it might have more than one range value associated with it. If we look close at the pairings we see that 1 is paired with -2 and then again with 1. This violates our definition of a function. Each of our values in the domain can only be assigned to one value in the range.
Before we move on to other representations of functions, let’s look one more set of ordered pairs.
We can list the domain of and the range of . It is important to notice that it is not a problem for several values in the domain to go to the same range value. This does not violate our definition of a function. So be careful!
Rather than using ordered pairs to describe functions and relations, a more common form is with the use of equations.
Usually, you will see a function written as something like . Many times teachers and books will use and y interchangeably. Recall that when graphing lines, we used the notation . Now we can see that this is a function because each value in the domain will be paired with only one range value. In this case, the domain consists of values that you can use for x. Sometimes domain is referred to as the input of a function because it is the values you are allowed to put in for x. Remember that domain referred to our independent variable.
Also recall that the range, or the y-values referred to the dependent variable. When you look at the function in its equation form, this makes sense. You get to choose the values for x that are put into the equation and then the y-values are dependent upon those x-values.
So here’s how so much confusion can arise from working with functions. Look at all the terminology we’ve used so far and the many different ways to describe the same thing.
Let’s go back and look at the function . A lot of times the notation of can be confusing. But f is just the name of the function and x is the independent variable in the function. When combined together we say that is the value of the function at x (again just another way of saying the y-value).
- Domain, independent variable, x, input
- Range, dependent variable, y, output, f(x)
We could name and label the same expression in a different way without changing the meaning. For example, we could call it . In this case, we have named the function s and used t is the independent variable.
In application problems, it is VERY common to see different labels for names and the independent variable. We’ll use some of these different notations in examples later in this lesson and also in the lesson evaluating functions (link to functions-evaluating.doc)
With so many terms and notations to work with we can’t forget to talk about the domain and range of . For the domain we want to know what values of x are allowed to be put into the function. In this case, we can put any value of x into the function. If we do that, our output or our range can also be any value. In this case, we say that the domain and the range are all the real numbers.
It turns out that any linear function will have a domain and a range of all the real numbers. In cases of radicals or fractions we will have to worry about the domain of those functions.
- In the function we will only be allowed to use positive values of x. Otherwise we have a negative value underneath the square root and we can’t take the square root of a negative number and still end up with a real number. When we evaluate the square root of a positive number, we will get a positive number as a result. (You may recall other times when it was acceptable to use a positive and negative in front of a square root symbol, but if we do that here, we will not have a function.) So for this function we have domain: and range: .
When working with radical functions, we can usually set what is underneath the radical greater than or equal to zero and solve the resulting inequality to find the domain. Some students like to look at a graph of a function to help them in determining the range.
- Consider the function . In this situation, as with any fraction, we have to be careful not to have zero in the denominator. So in this case, we want to make sure that . This means that . Any other value of x is allowed in this function, however we cannot use 2 because we will have zero in the denominator. We do not have any restrictions on the range, so we say domain: all real numbers except 2 and range: all real numbers.
When working with rational functions, set the denominator equal to zero and this will give you the value(s) that CANNOT be used in the domain.
One quick and easy way to determine is an expression
or a set of ordered pairs is a function
or not is to graph
it. All functions must pass what we call the vertical line test
. That means we must be able to draw a vertical line
anywhere on the graph
and have that vertical line
only touch in one place. If that happens, then the graph
is a function. If at any place on the graph
we are able to draw a vertical line
that touches the graph
in two or more places, then the graph
is not a function.
As mentioned earlier, graphing can also be used to determine domain and range of functions. You should try to graph an expression by hand or on your calculator to help you understand the vertical line test and domain and range.
A basic understanding of functions, terminology, and notation is necessary in order to be able to use functions in later lessons. Make sure you complete the following examples and understand their answers before going to any other lessons on functions.