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Whether you’re new to functions or just confused by them, this lesson covers the basic underlying concepts. Simple explanations and references to the real world make it easy to understand.

What Are Functions?

We use the word ‘function’ in everyday life to mean all kinds of different things. A function is what something does. You could say that the function of a cup is to hold water. A function can also be a special event. But in math, ‘function’ has a special meaning. A mathematical function defines a relationship between inputs and outputs.

The easiest way to get a handle on functions is to think of a function as being like a factory. Let’s say you want to make some paper. You would need:

  • Raw Materials: Paper is made from dead trees, so, for your paper factory, you’d need a steady supply of lumber.
  • Processing Equipment: You can’t just cut down the trees and get paper. You have to chop up the wood, grind it into a pulp, lay it flat, dry it out, bleach it to make it white, cut it into sheets. . .

The dead trees go into the factory, and they’re processed on the equipment. Then paper comes out of the factory and into your printer. The trees are the input, and the paper is the output. The specific kinds of equipment in the factory control the process of turning the input into the output.

A function is just like a factory, except the raw materials are numbers, and the processing is doing math with them.

A mathematical function gives you a process for transforming inputs into outputs, just like a factory gives you a process for transforming paper into trees. Where the processing equipment in a factory might be conveyor belts and driers, the processing equipment in a function is a rule for doing some kind of mathematical operation, something like this:

A function example

Rules for Functions

Think of the paper factory again. You can actually make paper out of all kinds of things: you can make it out of trees, or other plants, or make recycled paper out of other paper. But you can’t put just anything into the paper factory and expect to get paper at the end. If you tried putting footballs through your paper factory, you wouldn’t get very far!

Functions are the same way. You can’t put any number into any function.

  • The domain of a function is all the possible inputs.
  • The range of a function is all the possible outputs.

You’ll get into the reasons why the domain of a function can be restricted in another lesson; for now, just remember that it can.

Another important property of functions is that for every input to a given function, there is only one output. If we look at our example function again, if you plug in 3 and apply the rule ‘multiply by 2,’ you’ll only ever get 6 as the output. There’s no possible way to get any other number with 3 as the input. You can put in a different number and get a different output with the same rule, but with the ‘multiply by 2’ rule as our function, you can’t put in 3 and get 8, because 3 times 2 does not equal 8.

For a completely different function, you would get a different output for putting in 3 because you’re applying a different rule. But for any one function, each input can only have one output. If you can get multiple outputs from the same input, then the rule you have isn’t actually a function.

Function Notation

So far we’ve been using diagrams to show how functions work, but that’s not how they’ll actually appear in real life. Instead, you’ll see something called function notation. Function notation is the set of conventions that mathematicians use to write about functions. They don’t want to go around drawing diagrams all day, so everyone agreed on a different way to do it.

The confusing part about function notation is that it’s out of order. It doesn’t show the input first, the processing second, and the output last.

Here’s how function notation looks:

F(x) = x + 2

This function is named F.

X here is a variable representing any number in the domain of the function; any number that you could possibly plug in. The F with the x in parentheses is pronounced ‘F of x‘ and refers to the output that you get when you plug x into the function’s rule for processing numbers. ‘x + 2′ gives you the rule.

This kind of notation can be confusing because it looks like you’re supposed to multiply F times x. But that’s not true. F is not a variable; it has no value. It’s the name of the function. F(x) has nothing to do with multiplication!

If it looks really confusing right now, just stick with it. Function notation is hard for a lot of people at first, but once you use it more, you’ll get used to it.

Lesson Summary

In this lesson, you got an introduction to functions. A function is a rule for transforming a certain set of inputs into outputs. Think of it as being like a factory, where you put in some raw materials, put them through some processing, and end up with a finished product – only in a function, the ‘processing’ is actually math.

For each input into a function, there will be one, and only one, output. The domain of a function is all the possible inputs. The range of the function is all the possible outputs.

Function notation can be confusing, but it gets easier as you use it – promise! It might help you to write down the ‘translation’ shown here from function notation to the diagram; you can use that to solve problems until reading function notation becomes natural and easy.