What is Function?

What is a Function?

A function relates an input to an output.

It is like a machine that has an input and an output.

And the output is related somehow to the input.


                                  f(x)   f(x) = … ” is the classic way of writing a function.
And there are other ways, as you will see!

Input, Relationship, Output

We will see many ways to think about functions, but there are always three main parts:

  • The input
  • The relationship
  • The output

Example: “Multiply by 2” is a very simple function.

Here are the three parts:

Input Relationship Output
0 × 2 0
1 × 2 2
7 × 2 14
10 × 2 20

For an input of 50, what is the output?

Some Examples of Functions

  • x2 (squaring) is a function
  • x3+1 is also a function
  • Sine, Cosine  and Tangent are functions used in trigonometry
  • and there are lots more!


First, it is useful to give a function a name.

The most common name is “f“, but we can have other names like “g” … or even “marmalade” if we want.

But let’s use “f”:


We say “f of x equals x squared”

what goes into the function is put inside parentheses () after the name of the function:

So f(x) shows us the function is called “f“, and “x” goes in

And we usually see what a function does with the input:

f(x) = x2 shows us that function “f” takes “x” and squares it.

Example: with f(x) = x2:

  • an input of 4
  • becomes an output of 16.

In fact we can write f(4) = 16.


The “x” is Just a Place-Holder!

Don’t get too concerned about “x”, it is just there to show us where the input goes and what happens to it.

It could be anything!

So this function:

f(x) = 1 – x + x2

Is the same function as:

  • f(q) = 1 – q + q2
  • h(A) = 1 – A + A2
  • w(θ) = 1 – θ + θ2

The variable (x, q, A, etc.) is just there so we know where to put the values:

f(2) = 1 – 2 + 22 = 3


Sometimes There is No Function Name

Sometimes a function has no name, and we see something like:

y = x2

But there is still:

  • an input (x)
  • a relationship (squaring)
  • and an output (y)


At the top we said that a function was like a machine. But a function doesn’t really have belts or cogs or any moving parts – and it doesn’t actually destroy what we put into it!

A function relates an input to an output.

Saying “f(4) = 16” is like saying 4 is somehow related to 16. Or 4 → 16

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