Our starting point in this course is the mathematical idea of a function.

In this lecture, I'll talk about what functions are and how to think about them,

how to use functional notation, and how to interpret functions as graphs.

Let's get started.

In this course, I think the best way to think about functions are as an action,

which I've illustrated here. So a function is an action.

It takes an input, some number x as an input,

f, that's the name of the function, it does something to x,

and then it returns, it outputs a new value, which we call f(x).

So to specify a function we have to specify what this action is, what the action does.

So as a concrete example, let's say f is a function that triples its input.

So we could write an equation for that.

We could say f of x equals three x.

So f triples the input.

So suppose that our input was two.

Then it would output six, because two times three is six.

Using this notation, we'd say f of two, we replace x with two, it'd be two times... er... three times two is six.

So we'd say f of two, f acting on the number two, gives the output six.

Let's try another input.

Here's our function, call it f, let's say the input now is five,

the output would be fifteen.

Five times three. F of five - we take five as an input and then this tells us to do f to it, in this case that means multiply by three -

three times five equals fifteen.

Let's do one more quick example.

Same function f.

Suppose the input now is zero.

That's.. erm.. nothing

Zero times three is still zero, so in this case, it outputs the same number that it input.

If you have nothing, and you triple it you still have nothing.

Using this notation, f of zero is three times

zero, three times zero is zero.

So the rule is, x is the input

and then you evaluate the output on the right hand side here

by replacing whatever the input number is on the right.

So here we have a two, we replaced x with two.

Here we have a five, we replace x with five. If for example

we had f acting on some other variable,

the letter z, that would be just three times z

The actual value would depend on z.

If we had something like f acting on

a heart, who knows what that would mean, you have a heart, then you triple the heart,

you have three hearts.

Let's do another example.

This time the function, I'll call it g, will be

x squared plus one. So remember, a function is an action.

It takes a number as input, it does something to it, and it gives

an output. We specify the function by saying what that action is.

What rule does it follow to go from the input

to the output. In this case, the rule is: take the input number,

square it, and add one. So let's evaluate

this function for a few different values.

So, let's say our input is three.

So that would mean g of three.

So what do I do, I take three and I square it, that's the rule,

three squared plus one,

well, three squared, that's nine

nine plus one is ten. So we would say that

g of three is ten. We start with three, g acts on it, we

get ten. Using this notation, here's g,

three comes in, inside this box,

three gets squared and then gets one added to it,

so ten comes out. Alright, let's

do another example.

Suppose our input is zero.

So we start with zero and then we do what the function tells us to do

to evaluate it. The function says square it and add one.

So we've got zero squared and I add one

and zero squared, that's zero times zero,

zero times zero, well that's still zero,

So g of zero is one. We would have

one coming in the box - sorry, we would have zero coming in

the box and one coming out the box. Let's do one

more. Let's try a negative number now.

So our input is negative two.

That means we would take negative two,

square it and add one.

Negative two squared, well that's negative

two times negative

two. Negative two times negative two, the negatives cancel,

that would be plus four,

plus one. So that would be five.

One last thing I want to mention, just

a note about notation. So, notice I'm using

parentheses say here and here, but these parentheses

mean different things. Here, this means multiply. Take negative

two, multiply it by negative two. Here, this

doesn't mean take negative two and multiply it by g. This

means take the number negative two, view it as input to the function

or the action g and let g act on it.

So the notation can be a little bit confusing at first, particularly if you haven't seen this

before, but it's important to remember that a function is an action

and this type of notation, functional notation,

the parentheses is interpreted as the input value, it doesn't mean multiply.

I'll have more to say about functions in the next video.

But before you watch that video, I suggest that you try out the quiz

that immediately follows this video in the navigation bar.

A reminder that the quizzes are not part of your evaluation for this course.

In fact, the scores aren't even recorded at all. They're just a chance for you

to practice some of the ideas you're learning and to take a break from listening to me talk.

So, give it a try.