How do you represent functions?

There are many kinds of relations. Among the most important algebraic relations are functions. A function is a relation in which one variable specifies a single value of another variable. For example, when you toss a ball, each second that passes has one and only one corresponding height. Time only goes forward, and never repeats itself. The height of the ball depends on how much time has passed since it left your hand. This is a one way relationship-although each moment of time is unique, it is possible for the ball to be at a particular height more than once as it goes up and then down. Knowing the time will tell you the height, but knowing the height won’t give you an exact time.

The parts of a function are called inputs and outputs. An input is the independent, non-repeating quantity. The output quantity is the dependent quantity. The value of the output depends on the value of the input. For each input, there is a single output. In the case of tossing a ball in the air, time is the input and height is the output.

Let’s look at a few more examples to get comfortable recognizing what is a function and what is not. Remember the last time you were in a parking lot? You won’t be surprised to know that there’s a relation between the number of cars and the number of tires in a lot-the number of cars and the number of tires are linked. Is this relation a function? Can you use the number of cars to correctly figure out the number of tires?

Yes, you can. Every single car has 4 tires, so the number of tires depends on how many cars are in the parking lot. Every input of cars specifies a single possible output of tires. (In this example, the relation of tires to cars is also a function-the number of tires also specifies the number of cars.)

Now consider a different relation, between houses and the people who live in them. If an address is the input, and the output is the occupants, is this relation also a function? Think of your own house or apartment-are the people staying there always the same?

Nope. That time you went to camp, the occupancy changed. Every time you had a friend stay over, it changed again. Because a single address can produce more than one set of occupants, the relation is not a function.

Here’s a good rule of thumb to use to recognize functions: If you put the input in more than once, are you guaranteed to always get the same output? With the cars and wheels, the answer is yes. For an input of 25 cars we always get an output of 100 tires, no matter which 25 cars drive into that parking lot or when they arrive. The relation is a function.

With the houses and occupants, the input of an address is not guaranteed to always produce the same output, because a house stays put while people come and go. The relation is not a function.