What is a Function? A Simple Approach

There are many ways to understand what is a function. One way that is very useful is to consider it a rule, which given an input, gives an output. This way of looking at functions is very useful in calculus, and we'll be using it almost always.

What do I mean by a rule which asigns an output to an input? Consider a box, like the one below:

Box representation of a function

The box receives an "x" and gives the output "f(x)".

The functions we use in calculus are rules that relate numbers. So, in our case, the "x" in the box represents any real number.

I'll give you an example. Let's make x equal to 1. Then,

Box function with x=1

In this case, we say that the function f(x) is evaluated at 1. We read f(1) "f of 1".

An actual function may look like this:

An actual function in a box

If we make x equal to 1, we have:

An actual function in a box with x=1

Me Example

You may already have an idea of what is a function after taking algebra. When thinking about functions you probably think only about numbers.

Don't think that functions are only about numbers. In fact, almost anything can be considered a function!

Here are some hilarious examples of functions. I saw Salman Khan from the KhanAcademy teach using them, and I found them very useful.

Let's say that I am a function. Let's study the Pablo function. In case you don't know, Pablo is my name.

What will happen if you give the Pablo function the input "food"? Let's see:

Pablo(food) = calculus pages

If you give me food, I'll output calculus pages!

Now, let's consider the "you" function. Let's give "you" the input "calculus pages":

You(calculus pages) = A's on your exams

You output A's on your exams.

Now, let's see what this expression means:

You(Pablo(food)) = ?

This may seem convoluted, but is very simple. Let's replace Pablo(food) by its equal, "calculus pages":

You(Pablo(food)) = You(calculus pages) = A's on your exams

You may or may not find these examples useful. I found them hilarious when I saw them.

They are useful to show what is a function and that they are not limited to relationships between numbers.

Solved Problem 1

Let's do some function problems. Let's say that:

Example function

What is f(0)? Simple, right?

Evaluating the function at x=0

What is f(-1)?

Evaluating the function at x=-1

Solved Problem 2

Now, let's a do a slightly more complicated problem. Until now, to design a function we've been using only the letter f, as in f(x).

When we have more than one function, we need other letters to name the functions. The favorites are "g" and "h".

In this problem we have three functions:

Definitions of the three functions

The question is:

The unknown

Looks complicated, doesn't it?

Let's tackle it by parts. First, let's do g(3):

Evaluating g(x) at x=3

Now, we need f(0) to have g(3) complete. Let's find that:

Evaluating f(x) at x=0

Putting f(0) into equation (1) we have:

Evaluating g(3)

And finally:

Final answer to second example

Solved Problem 3

Here we have another interesting problem:

Third example of functions problems

Let's think. What is f(x)? It is 1/(1+x). So,

Answer to first question

Now, let's find this:

Second unknown

After doing the first one, this is easy:

Answer to second question


Here's a little exercise for you:



  • There are many ways to understand what is a function, because it is a very general concept.
  • One way that is very useful is as a box, which given an input, gives an output.
  • Do some function problems!

To learn more about functions:

Return to Mathematical Functions

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