A Rule, and the Picture of It

A function is a rule that answers one number for each number given. The graph is a record of what the rule did, and a table of answers never determines the rule.

Most of the trouble people have with calculus is not with calculus. It is with

having been taught that a function is an expression with an x in it, and then being

asked to do things to functions that only make sense if a function is something

else.

So it is worth being slow here for one lesson. A function is a rule. You hand it a

number, it hands one back, and it hands back the same one every time you ask.

FIG 1A rule, written out

: a name for the rule itself, so that it can be talked about without being carried around in full. This is the object the course does things to
: a placeholder for whatever number is handed in. It has no value of its own, and the rule is unchanged if every x in it is renamed
: becomes. It separates what arrives from what is handed back, and it is the arrow that makes this a statement about a rule rather than an equation to solve
: the answer, written as a recipe in terms of what arrived. A recipe is the most convenient way to specify a rule, and it is not the only way

Read the arrow as the word becomes. This says: the rule called f is the one that turns any number given to it into twice that number plus three. It does not say that f equals anything, and it does not say what x is. The letter x here is a placeholder for whatever arrives, which is why it can be renamed throughout without changing the rule at all.

The rule as a machine

The recipe above has two stages, and it is worth watching a number go through them,

because the stages are the thing the next lessons will take apart.

FIG 2Five numbers handed to the same rule

step	handed in	after doubling	after adding three	what happened
1	0	0	3	The rule answers three when given nothing to work with.
2	1	2	5	One in, five out.
3	2	4	7	Each extra one on the way in adds two on the way out, every time.
4	3	6	9	Still two. That steadiness is the only reason this rule is called linear.
5	-2	-4	-1	Negative numbers are handed in the same way and get the same treatment.

5 steps

The middle column is not part of the answer, it is part of the working, and it is shown because the next lesson will care very much about what happens inside a rule rather than only at its ends. Notice the step from row to row in the last column: it is two every time, regardless of where you start. Almost no rule behaves like that, and the whole of the next lesson exists because most do not.

The picture, and what it is a picture of

To draw the rule you ask it for an answer at a great many inputs and mark each answer

as a height above the place where it was asked. That is all a graph is: a record of

a large number of separate questions, with a line drawn through the answers by hand.

The line between two marks was drawn by the person plotting, not by the rule. Usually

that is harmless. Here is when it is not.

FIG 3Two rules that agree at every whole number

the first rulethe second rule

Check these two at zero, one, two, three and four and they agree exactly, because the extra piece in the second rule is zero at every whole number. Check anywhere else and they do not. No amount of further checking at whole numbers would ever separate them, and a third rule could be built to agree at any larger set of points you cared to name. A table of answers is evidence about a rule; it is never a specification of one.

That is a practical warning as much as a mathematical one. Any time a curve is fitted

to measurements, the fit is chosen out of a family of candidate rules decided in

advance, and the data did not choose that family. The data cannot. Everything the fit

says between the measurements comes from the choice of family, which is why the honest

question about a fitted curve is never how well it fits but what it was allowed to be.

A letter that picks the rule

One more distinction before any calculus. In an expression with several letters, some

are inputs and some are selectors.

FIG 4One knob, a family of rules

which member of the familya = 1

the rule picked by the knob

Both letters in the expression are doing completely different jobs. Moving along the curve varies x, which is the input the rule acts on. Moving the knob varies a, which does not feed into any calculation but chooses which rule is being drawn. At a of zero the family member is a flat line, and notice that the choice flips the curve over as the knob passes through zero. Nothing in the algebra announces which letter is which: that has to be said in words, and in this course it always will be.

What to take forward

Three things, and they are small.

A function is a rule that answers exactly once. A graph is a record of what the rule

answered, with the gaps filled in by the draughtsman. And no finite set of answers

pins a rule down, which is why every statement in the rest of this course is about the

rule itself rather than about a list of its values.

The next lesson asks the first real question. Given a rule, how fast is its answer

changing at one particular place? The rule in the trace above changes by exactly two

per step wherever you look, which makes it a bad example, and that is exactly why the

next lesson starts somewhere else.

Recap

A function is not a formula. It is a rule that takes one number and answers with exactly one number, and the formula is only the most convenient of the many ways to write such a rule down. A lookup table, a procedure, a measured record and an algebraic expression are all functions if they answer exactly once.
The graph is a record of what the rule answered, not the rule itself. Reading a graph is therefore reading history, and the only things that can be read off it honestly are things the rule actually did somewhere.
No finite collection of answers picks out a rule. Two rules can agree at every point you check and differ everywhere between, so a rule is only pinned down by a statement that covers every input, which is why the rest of the course works with the statement and not with samples.

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

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