Before diving into calculus,

we should first talk briefly about what functions are and where we use them.

Essentially, a function is a relationship between some inputs and an output.

So, for example, if I had a function for

modeling the distribution of temperature in this room,

I might input the x, y, and z coordinates

of a specific location I'm interested in as well as the time, t.

And then the function would return me

the temperature at that specific point in space at that moment in time.

Like so many areas of maths,

even if the idea is quite straightforward, often,

the notation can make things unnecessarily confusing.

We'll be confronted by this again later in the course and although it's

sometimes fairly arbitrary historical reasons that decide this,

like different people inventing different parts of maths at different times,

it can also be because a particular notation style is

more convenient for the specific application it was developed for.

However, and here is where a lot of the problems seem to arise,

in order to use and play with the interesting applications of maths,

it requires you to have done quite a large amount of often quite boring groundwork.

Mathematical language is like learning any other language in that respect.

You can't enjoy French poetry until you've learned a lot of

French vocabulary and grammar including all its quirks and irregularities.

Quite understandably, some people find this off-putting.

My French is still terrible, for example.

This is made worse by the fact that most people have not even

realised that there is maths poetry waiting for them at the end of this algebra tunnel.

Machine learning is a whole genre of this poetry,

so stick with us for the rest of the specialisation and you'll

be ready to appreciate it and even write some of your own.

Back to functions.

We often see expressions such as f of x equals x squared plus three.

Perhaps, the only thing worth mentioning about this notation style is,

yes, I admit it,

it is absurd that you should somehow just know that "f(x)"

means f is a function of x and not f multiplied by x.

Sometimes, this gets genuinely unclear when

other bracket terms appear in your expression.

For example, f of x equals this thing over here.

You can assume that g,

h, and a are all not variables.

Otherwise, I would have to write f of x,

g, h, and a.

But you could only know for sure what was going

on here if it was explained to you with more context.

For example, is g a function being applied to x?

What about h and a over here?

Maybe they're both just constants,

although you do face the same problem of missing context in any language like

the famous panda who goes to a restaurant and eats

shoots and leaves and they were both delicious.

I can only apologize on behalf of maths and hopefully console you with two facts.

Firstly, maths is at the core of my day job and I often still get confused myself.

And secondly, I still love it and I get better at

second guessing any missing context all the time,

just like any other language.

In the following exercise,

I'm either going to show you some data or describe a concept to you

and you're going to have to select a function

which you think might be used to represent it.

This is not intended to be a difficult exercise but what I would like you to

understand is that selecting a function is the creative essence of science.

This process of selecting a candidate function or hypothesis to

model a world is what the great geniuses of science are remembered for.

There then follows a potentially long and difficult process of testing

this hypothesis but there will be nothing to test without that first creative step.

Calculus is simply the study of how these functions change with

respect to their input variables and it allows you to investigate and manipulate them.

But ultimately, it's just a set of tools.

And by the end of this course,

you'll be using them yourself to model some real world data.