Something In, Something Out

Last timeA Letter Is Standing In For Something

Function notation is a machine with labelled inlets. Reading it is a matter of knowing which letters go in, which are held still, and what comes out.

The previous lesson took an inventory of the letters in a formula. This lesson

is about the brackets, which is where most of the apparent complication in

modern notation lives.

The shape to read is $f(x)$ . There is a name, a bracket, and something inside

it. The name is a machine, the thing inside is what is being fed in, and the

whole expression stands for what comes out. That is all, and it is worth saying

plainly because the notation is almost identical to multiplication and gets

confused with it daily.

A name, an input, and a value

The first thing to be clear about is that $f(x)$ is a noun and not a verb. It

does not mean do $f$ to $x$ . It denotes the single thing that comes out, and

that thing can then be added, compared, substituted and summed exactly as any

other quantity can. When you see $f(x) + 3$ , the machine has already run and

its output is having three added to it.

The second thing is that the bracket is doing the job of the word of. In

$f(x+1)$ , the quantity $x+1$ is being fed in. In $w(x+1)$ , where $w$ is an

ordinary number, the very same arrangement of marks means multiply. There is no

way to tell these apart from the symbols, which is uncomfortable and true. The

only thing that distinguishes them is whether $f$ or $w$ was introduced as a

function or as a number, which sends you back to the declarations from the

first lesson.

FIG 1One machine, drawn

Two inlets, one outlet. Every piece of notation in this lesson is about which quantities arrive at which inlet, and the only genuinely hard part is that one of the inlets is routinely left out of the writing.

What moves and what was fixed

Now the line that matters, because it is the shape of nearly every formula in

machine learning.

The lesson stops here

7 more paragraphs to go

You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.

The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.

See the plan The contents

This is the reading half

Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.

The contents

Something In, Something Out

A name, an input, and a value

What moves and what was fixed

7 more paragraphs to go

This is the reading half

Read alongside