The Function

Most people meet a function as a formula with an f in front of it. That description is too loose. It lets in rules that mathematics does not count as functions, and the ones it wrongly lets in are exactly the ones that break everything built later.

A function A rule that assigns to every input it accepts exactly one output. It has three parts: the set of inputs it accepts (its domain), the set the outputs are drawn from, and the assignment itself. "Exactly one" is the working clause: y² = x is not a function of x because x = 4 gives both 2 and −2, while √x is one because the root sign is defined to mean the non-negative root only. Written f(x), where f names the rule and x the input, so f(4) is the output at 4 and there is no multiplication in the notation. is a rule with three parts: a set of inputs it accepts, a place the outputs live, and an assignment giving every accepted input exactly one output. The words to hold on to are "exactly one".

Feed the machine the same number twice. The same thing comes out twice. A rule that returned 7 on the first run and 8 on the second would not be a function, however respectable its formula looked, because the input no longer decides the output.

Now the rule that fails. Take y² = x. Ask it for x = 4 and it has two answers, y = 2 and y = −2, because both of those square to 4. Nothing in the rule chooses between them. So y² = x is not a function of x. It is a usable equation and it describes a real curve. It is not a function.

This is repairable, and the repair is where a convention you have already used comes from. The square root sign √ is defined to mean the non-negative root only, so √4 is 2 and never −2. That definition exists to turn a two-answer rule into a one-answer one. Half the curve is thrown away to buy the "exactly one".

One note on notation, because it trips people who read it as algebra. In f(x), the letter f names the rule and x names the input, and f(4) means the output that rule gives when fed 4. There is no multiplication in it.

A rule can be written down and still have numbers it cannot take. Handing it one of those is not a mistake that produces a bad answer. It produces no answer.

The domain The set of inputs a function accepts. It is part of the function rather than a note attached to it, so two rules with the same formula and different domains are different functions: 1/x excludes the single point 0 and √x excludes every input below 0, which are two different shapes of exclusion. Sine and cosine accept every real number. Tangent and secant exclude the angles where cosine is zero, and cotangent and cosecant exclude the angles where sine is zero, because a division by zero has no value there. of a function is the set of inputs it accepts. It is part of the function, not a footnote to it. Two rules with the same formula and different domains are two different functions, and the last step of this page turns on exactly that.

Two refusals matter more than all the others. The first is division by zero. In 1/x, the input 0 is refused, because no number multiplied by 0 gives 1, so the expression 1/0 names nothing. It is not large and it is not zero. There is no number there to name.

The second is the square root of a negative. In √x, every input below 0 is refused, because no real number multiplied by itself is negative. Complex numbers do give √(−4) a value, and they are a different subject from this page. Here the answer is that the rule refuses.

Those two refusals have different shapes, and the shape is worth seeing. 1/x punches out a single point and keeps everything on both sides of it. √x cuts off a whole half of the line at once and keeps the rest. A domain is not always a formula with a hole in it, and it is not always an interval.

One honest warning about the picture. 1/x has no ceiling near zero. At x = 0.1 it is 10, at x = 0.01 it is 100, at x = 0.001 it is 1000, and the numbers keep going for as long as you keep dividing. No drawing can show all of that, so whatever the figure does when the output leaves the top of its scale, that is the drawing running out of room and not the machine returning a finite answer.

The mark for an excluded point has a name and it is used from here on. An open circle A small unfilled ring drawn on a number line or a curve to mark a point that is not included: the position is shown so you can see where it would be, and the ring is left hollow so you can see that no value sits there. It is the standard mark for an input outside the domain, such as x = 1 on the graph of (x² − 1)/(x − 1), which is the line x + 1 with that one point removed. A filled dot means the opposite, that the point is included. drawn on a line or a curve says the point sits there and is not included. It is drawn so you can see where it would be, and left unfilled so you can see that it is not there. The last step of this page comes back to it, and a later page in this series is built on almost nothing else.

Here is a job with no f(x) in it. A beam sits on a pivot. A 12 kg block is bolted to the beam one metre to the left of the pivot and cannot be moved. You have a stack of one kilogram blocks and the whole right arm to put them on. How heavy does your stack need to be?

The answer depends on where you stand it, and the mechanism is old. A beam balances when the turning effect on one side matches the turning effect on the other, and the turning effect of a weight is its mass multiplied by its distance from the pivot. The left side gives 12 · 1.0 = 12, fixed. So your stack, at distance d, must satisfy m · d = 12, which rearranges to m = 12/d. That is the rule from the previous step, scaled by 12, and it now has a unit attached to it.

Two calculations run in this figure and neither one reads the other. The line above the drawing works out 12/d from the rule and the distance, and nothing else. The tilt of the beam is worked out from the mass and the distance of the blocks that are actually on it, and nothing else. They agree by coming out the same, not by being the same sum written twice, which is what makes the beam going level a check on the rule rather than a picture of it.

Now walk the stack in towards the pivot. At 2.0 m the rule asks for 6 kg. At 1.0 m it asks for 12. At 0.5 m it asks for 24, which is every block the drawing has. At 0.2 m it asks for 60, and you cannot build that here. It would be easy to conclude that you need a bigger machine, and for every distance above zero that is true: the number exists and something could be stacked that high. At exactly the pivot the conclusion breaks. Distance 0 means turning effect 0, whatever the mass is, so twenty-four blocks on the pivot turn the beam exactly as much as no blocks do. Nothing levels it. That is the difference between a number too big to draw and no number at all, and it is what 12/0 being undefined means with your hands on it.

The domain answers which inputs go in. It says nothing about what comes out, and those are separate questions with separate answers.

The range The set of outputs a function produces. It is what the rule does produce as the input runs over the whole domain rather than what it is permitted to produce, and it is usually harder to pin down than the domain: x² accepts every real number and yet produces nothing below 0. Sine and cosine have range [-1, 1], capped by the radius of the unit circle. Tangent and cotangent range over every real number. Secant and cosecant take every value of magnitude at least 1 and nothing strictly between -1 and 1. of a function is the set of values it produces as the input runs over the whole domain. Not the values it is allowed to produce, the ones it does produce.

Take x², whose domain is every real number, so nothing is refused at the input end at all. Squaring a positive gives a positive. Squaring a negative also gives a positive, because two negatives multiplied give a positive. Squaring zero gives zero. So no matter how far you sweep the input in either direction, the output never lands below 0. The whole negative half of the output line is unreachable, and it stays empty because of what the rule does, not because you have not swept far enough.

That is the range being found rather than stated, and it is worth noticing that the input side gave no hint of it. A rule that accepts everything can still produce almost nothing.

The range is usually harder to pin down than the domain, and pretending otherwise helps nobody. Finding a domain is a search for local failures: a zero on the bottom of a fraction, a negative under a root. Finding a range is a question about the rule as a whole, and for many rules it takes real work.

Two more, for contrast. 1/x refuses the input 0, and it also never outputs 0, because a fraction with 1 on top is never zero however large the bottom gets. Every other number does come out. And 2x + 1 refuses no input and misses no output: give it any target y and the input (y − 1)/2 hits it. That last case matters more than it looks. A page this full of refusals leaves a reader assuming every rule has one, and this one has none at either end.

Here is the misreading this step exists to remove. A graph is not a drawing of a formula, and the curve is not an illustration of the algebra. The graph is a record.

For every input the function accepts, take the pair made of that input and its output. Mark the point that far along and that far up. The graph is the set of all those points and it contains nothing else. A curve looks continuous because the points are dense, not because a line was drawn through a formula.

The two number lines you swept in the previous step already hold every pair. They hold them awkwardly. To see how the output changes as the input moves, you have to follow the far end of a moving arrow across a gap. Standing the output line up at right angles turns each arrow into one point, and a moving input then leaves a path you can take in at once. The fold adds no information and loses none. It is the same set of pairs, filed differently.

With the plane built, the test people are given first falls out on its own. The vertical line test A check on whether a drawing in the plane is the graph of a function of x. Place a vertical line anywhere: it selects one input and every output that input has, so if it meets the drawing more than once, that input has more than one output and the drawing is not a graph of a function. It is the definition read off the page rather than an extra rule. There is no matching horizontal test, because many inputs sharing one output is allowed.: put a vertical line anywhere on the plane. That line picks out one input value and every output that input has. If the line meets the drawing more than once, that input has more than one output, and the drawing is not the graph of a function.

Bring back y² = x from the machine. Its drawing is a sideways curve, and a vertical line to the right of 0 cuts it twice, at y and at −y. That is the machine forking, drawn. The test is not an extra rule to remember. It is the definition read off the page.

Horizontal lines are a different matter, and nothing stops one crossing a curve many times. Many inputs sharing one output is ordinary and allowed. One input with many outputs is what the definition bans. The next step is what that asymmetry feels like from the inside.

Somebody hands you a plotted curve and no formula: a calibration chart for a sensor, a figure in a paper whose data table was not published. The procedure for getting the output at one input is two moves. Go up from your input on the horizontal axis until you meet the curve, then go across to the vertical axis and read the value where you land. Up finds the pair, across reports it.

Now run it backwards, because that is where the definition shows its shape. Start at a value on the output axis, go across to the curve, then down. You are asking a different question: which inputs produce this output?

The backwards question can have more than one answer. For x², the output 9 comes from x = 3 and from x = −3, so going across from 9 meets the curve twice and drops to two different inputs. It can also have none: no input makes x² equal −1, so going across from −1 never meets the curve at all.

Many inputs to one output is allowed. One input to many outputs is not. That is the asymmetry the whole definition rests on, and it is why the test that fell out of the fold is about vertical lines and there is no matching test for horizontal ones.

The domain step had refusals you could see coming. 1/x runs away near zero and √x cuts off half the line. This one is quieter.

Take the rule (x² − 1)/(x − 1). The bottom is 0 when x = 1, so 1 is refused and is not in the domain. Nothing new so far.

Now factor the top. x² − 1 is (x − 1)(x + 1). For every input other than 1, the factor of x − 1 appears on the top and on the bottom and cancels, so the rule returns x + 1. Its graph is the straight line of x + 1 with a single point taken out of it, marked with the open circle from the domain step.

Compare the two kinds of refusal directly. Near 0, the values of 1/x have no settled behaviour at all: they climb without bound on one side and fall without bound on the other. Here, neither side runs away. At x = 0.9 the rule gives 1.9. At x = 0.99 it gives 1.99. At x = 1.01 it gives 2.01. The values sit close together on both sides and get closer as the input moves in. At exactly x = 1 the rule gives nothing.

You could write down a second rule, x + 1, that refuses nothing. It agrees with this one at every input this one accepts. It is not the same function, because a function includes its domain and these two do not have the same domain. One of them has a value at 1 and the other has an input it turns away.

So stop here with the question, because it is a better thing to leave with than an answer. The values on both sides crowd towards something. The rule itself has nothing at the point they crowd towards. Whether that gap can be filled, what it would be filled with, and by what right anyone would fill it, are not questions this page can settle. A later page in this series is built entirely on them.

Every number on this page is worked out from the controls as you set them, including the ones in the legends and in the table. Nothing here was stored as a picture. Each figure keeps its own input and its own choice of rule, because the rules on offer differ from one figure to the next.