1.3 Inputs, Outputs, and What Makes a Function

Learning Objectives

By the end of this section, you will be able to:

Sylvia has a small pool full of water. She needs to empty it, scrub it out, and refill it before a pool party. She drains part of it with a hose, hauls some out with a single bucket, gets her two friends to help with three buckets at once, takes a break, cleans the empty pool, and then refills it with the hose. Only one of those activities happens at a time.

If you sketch the height of the water in that pool against the time since she started, you get a picture of the whole afternoon. The graph falls while she is emptying, falls three times as fast while three people are bucketing, sits flat at zero while she cleans, sits flat somewhere above zero while she takes a break, and rises while she refills. Somebody who has never seen the pool can read the story off the graph.

That is what this chapter is about. Two quantities are tied together — the time and the water level — and the tie between them carries information. Section 1.2 built the same kind of tie out of a growing pattern: the step number and the number of tiles. This section gives that relationship its proper name and its precise rules.

The name is function, and the precision is the point. A great many relationships between two quantities exist in the world, and only some of them are functions. The ones that are behave predictably enough to be worth writing down, graphing, and calculating with; the ones that are not, are not. Learning exactly where the line falls — and it is a sharper line than most people expect — is the single most important thing in this chapter.

The other thing this section delivers is notation. Once you are confident that a relationship is a function, you will want a short way to say "the water depth 40 minutes in." Writing a sentence every time is unbearable. Function notation compresses that sentence into four characters, and by the end of this section you will read \(d(40)\) as fluently as you read a sentence.

1.3.1 Input and Output

Two quantities, and one of them depends on the other

Start with the pool. Two quantities are moving: the time since Sylvia started, and the depth of the water. They are not on equal footing. The clock runs on its own — Sylvia's bucket does not speed it up or slow it down. The water depth, on the other hand, is entirely at the mercy of what Sylvia is doing at that moment, and when that moment is.

So there is a direction to the relationship. Time drives; depth follows. That direction is not a detail you can leave out and sort later. It is the first thing to nail down about any pair of quantities, because everything else in this chapter — the test for a function, the graph, the notation — is built on top of knowing which quantity is doing the determining.

Definition 1.3.2 — Independent and dependent variables: four words, two ideas, and one question sorts every row.

The dependent variable is another name for the output. It is called dependent because its value depends on the input.

Four words, two ideas. Input and independent variable are the same thing. Output and dependent variable are the same thing. If a question asks for the independent variable and you have been calling it the input all along, you already have the answer — just say it in the other language.

Which quantity is which. Read the middle column as "what you pick" and the right column as "what you get":

Every row of that table passes the same test. Ask yourself which quantity you could set on purpose, and which one you would then have to go measure. You can decide to pump 8 gallons; you cannot decide what the pump will charge you. You can pick the second on the stopwatch; you cannot pick where the drone will be.

The words that tell you which is which

Textbooks and teachers signal the input in at least four different ways, and the wording is the only clue you get. All four of these describe the same relationship:

• distance versus time
• distance with respect to time
• distance over time
• distance dependent on time

In every one of them, distance is the output and time is the input. The pattern is worth memorizing directly: the output is named first, the input is named after the connecting phrase. "Distance versus time" means distance depends on time, not the other way around.

Reading the four phrasings. Each row names the output first and the input after the connecting phrase:

Do not let the wording become the obstacle. Find the two quantities first, then ask the plain-English question: which one is doing the determining? The phrase is only there to tell you which order the author had in mind.

Order matters, and reversing it can change everything

Here is the fact that makes all of this worth the trouble. Swapping the input and the output gives you a different relationship, and the new one may behave completely differently from the old one.

Consider a person's name and their Social Security number.

• Name as input, Social Security number as output. You hand over the name "David Smith" and ask for the number. There are many people named David Smith, each with a different number. The relationship hands back several answers.
• Social Security number as input, name as output. You hand over a specific number and ask for the person. A Social Security number belongs to exactly one person. The relationship hands back one answer.

Same two quantities. Opposite behavior. The next subsection makes that difference official.

1.3.2 What Makes a Relationship a Function

The definition

Everything in §1.3.1 was setup. Now comes the rule that separates the relationships worth calculating with from the ones that are not, and it is short enough to memorize in one reading.

The words domain and range are the subject of §1.4. For now, read domain as "the collection of allowed inputs" and range as "the collection of outputs that actually come out."

The misconception that has to die first

Almost everyone, on first meeting the definition, remembers it as "no repeats." That is half right, and the wrong half is the half people apply.

Say it as a pair of one-liners:

• Two different inputs sharing an output: fine. Nothing in the definition forbids it.
• One input with two different outputs: fatal. That is exactly what the definition forbids.

Why is the asymmetry there? Because a function is supposed to answer a question. You feed it an input and it must come back with an answer — one answer, so you know what to write down. If the same input can produce a 5 sometimes and a 9 other times, the relationship has told you nothing. But if two different inputs happen to produce the same 5, there is no confusion at all: ask about either input, get 5, write down 5.

Here is a table where the repeats are right in front of you. Two continuous functions \(f\) and \(g\) are given by their values:

Look at the \(f(x)\) column: the output \(0\) appears twice, at \(x = 1\) and at \(x = 2\). \(f\) is still perfectly a function, because reading down the table no single \(x\) value has two entries beside it.

The test, stated as a procedure

Whatever form the relationship arrives in, the test is one question asked once per input, and a single input with two different outputs settles it. That input-with-two-outputs has a name, and it is a name you will use for the rest of your mathematical life.

Two rules, side by side

The clearest illustration uses two rules built from the same pair of quantities.

Rule G takes a person's name as its input and gives their high-school graduation year as its output.

Rule P takes a high-school graduation year as its input and gives a person who graduated that year as its output.

The two tables look equally tidy, and that is the trap. A short list of pairs can always be written down neatly; what matters is what the rule does when you feed it an input that is not on the list. The test asks about every input, not about the four the author happened to print. So run your own name and your own graduation year through both rules and watch them come apart.

1.3.3 Testing Every Representation

A relationship can arrive in five different costumes. The test never changes; only what you are looking at does. Learn what the fatal pattern looks like in each costume and you can decide any of them at a glance.

A set of ordered pairs

In an ordered pair \((x, y)\), the first number is the input and the second is the output. So the test reads: does any first number appear twice with two different second numbers?

Consider this set:

List the first numbers: \(-7, 3, 8, -6, -2\). All five are different, so no input is asked to produce two outputs. This is a function. The output \(5\) appears twice — once from the input \(3\), once from the input \(-6\) — and as established, that is allowed and irrelevant.

Now this set:

List the first numbers: \(9, 0, 4, 5, 2, 0, 3\). The input \(0\) appears twice, paired once with \(4\) and once with \(-3\). Ask this relationship "what is the output when the input is 0?" and it gives two contradictory answers. This is not a function, and the pair \((0, 4)\) together with \((0, -3)\) is the counterexample that proves it.

And this one:

Every first number is different. A function.

A table of values

A table is a set of ordered pairs written vertically. So the test is identical: does any value in the input column appear twice with two different outputs beside it?

Inputs \(1, 2, 3, 4\), all different. A function. The outputs grow wildly, which is not a problem — nothing in the definition says outputs must be well behaved.

Inputs \(0\) through \(6\), each once. A function — and again the output \(2\) appears twice, at \(x = 0\) and \(x = 3\), which changes nothing.

One trap worth naming. Reading a table sideways rather than down will fool you. The question is never "does the output column repeat," and it is never "does the pattern look nice." It is only ever "does one input have two outputs."

A mapping diagram

A mapping diagram draws the inputs in one oval, the outputs in another, and an arrow from each input to the output it produces. The test becomes visual and almost trivial: does any input have two arrows leaving it?

Suppose the inputs are \(\{1,\, 2,\, 3,\, 4\}\) and the arrows go like this:

Every input has exactly one arrow leaving it. A function. Two arrows arrive at \(7\) — one from \(1\), one from \(3\) — but arrows arriving at the same place is the repeated-output situation, which is fine. Arrows leaving the same place is the fatal one.

Change one line:

Now the input \(2\) has two arrows leaving it, headed to \(9\) and to \(5\). Not a function.

The picture is worth carrying: many arrows into one output is fine; two arrows out of one input is not.

A graph, and the vertical line test

A graph is the same information again, plotted. The standards say it directly: the graph of \(f\) is the graph of the equation \(y = f(x)\), so every point on the graph is a pair (input, output) — the \(x\)-coordinate is the input, the \(y\)-coordinate is the output.

That single sentence gives the test immediately. Two points with the same \(x\) and different \(y\) are exactly one input with two outputs. On a graph, two points with the same \(x\)-coordinate sit directly above and below one another — on the same vertical line. So the fatal pattern, drawn, is a vertical line that hits the graph twice, and that is worth stating as a rule of its own.

A circle is the same failure in a friendlier shape. On the circle of radius 5 centered at the origin, the vertical line at \(x = 3\) enters at the bottom and exits at the top, meeting it at \((3, -4)\) and \((3, 4)\), since \(3^2 + 4^2 = 9 + 16 = 25 = 5^2\). One input, two outputs. A circle drawn in the coordinate plane is a relationship between \(x\) and \(y\), but it is not a function.

A situation described in words

Here there is no picture to inspect, so you have to do it by reasoning. Name the two quantities, decide which is the input, then ask: could one value of the input honestly correspond to two different values of the output?

Sylvia's pool: at any single moment, the water in the pool has one depth. It cannot be 14 inches deep and 3 inches deep at the same instant. Every input of time has exactly one output of depth, so water depth is a function of time.

That reasoning survives everything the story throws at it. The depth goes down, then up, sits flat during the break, and drops in visible jumps as buckets come out. None of that matters. At each time, there is one depth.

Now compare a relationship that fails. The length of fence needed, with respect to the area to be enclosed. The input is an area; the output is a fence length. Take an area of 36 square feet. A \(6 \times 6\) rectangle encloses it with 24 feet of fence. A \(4 \times 9\) rectangle encloses the same 36 square feet with 26 feet of fence. A \(2 \times 18\) rectangle needs 40 feet. One input, three different outputs — and there are infinitely many more. Not a function.

One more situation, from a different afternoon than Sylvia's. Priya, Jamal and Wen spent a lazy 120-minute afternoon floating down a river in inner tubes. Priya kept noticing the water was deeper in some stretches than others; Jamal, who likes a number attached to everything, logged the group's total distance travelled on a GPS every ten minutes; Wen just kept saying "math is everywhere" and turned out to be right.

Jamal's distance-travelled graph is a function of time for the same reason the pool's depth was: at any one moment the group has covered exactly one total distance, so no single time can honestly own two different distances. But it does work the pool never did. Sylvia's depth went down, then up, then flat, then down again — the same depth turned up at several different times. A river current does not run backward, and neither does a group floating in it: once they pass a stretch of river, that stretch is behind them for good. So the total distance travelled only ever climbs, from 0 feet at the start to about 15,000 feet by the end of the float, and it is strictly increasing — no output on that graph ever repeats.

That is the question §1.3.4 takes up next, and it turns out to decide far more than it looks like it should.

1.3.4 A Catalogue of Relationships

The fastest way to sharpen the definition is to run it across a pile of unrelated situations and watch where it bites. Work down this list. For each one, name the two quantities, decide which is the input, and then decide.

Two more entries stretch the definition in useful ways, and both are worth knowing before §1.4, because each one is really a statement about which inputs a function is allowed to take.

A sequence is a function. You already built one, in §1.2's Growing Dots: \(D(0) = 1\), and \(D(t) = D(t - 1) + 4\) for every whole minute after that, giving \(1, 5, 9, 13, \dots\). Written out directly,

Check it: \(D(0) = 4(0) + 1 = 1\) and \(D(1) = 4(1) + 1 = 5\). This is a function. The input is the number of minutes and the output is the dot count, and minute 6 has exactly one value: \(D(6) = 25\). What makes sequences look different from the other functions here is that the input can only be a whole number — you cannot ask how many dots there were at minute 2.5, because the picture was only ever drawn one whole minute at a time. The allowed inputs are a subset of the integers, and §1.4 will call that its domain.

Functions do not have to be described by one formula. Define a rule this way: if \(x\) is a rational number then the output is \(1\), and if \(x\) is an irrational number then the output is \(0\). Every real number is either rational or irrational and never both, so every input receives exactly one output. It is a function, even though its graph is impossible to draw and its rule takes two lines to state. This is a useful corrective: a function is a promise about inputs and outputs, not a promise about formulas or smooth curves.

1.3.5 Function Notation

Naming the function

Once you know a relationship is a function, you need a compact way to talk about it. Writing "the depth of the water in Sylvia's pool 40 minutes after she started" every time is not sustainable.

Give the function a letter. Then attach the input in parentheses.

Definition 1.3.6 — Function notation: f names the rule, x names the input, and f(x) names the output.

The letters \(f\), \(g\), and \(h\) are the usual choices, the way \(a\), \(b\), and \(c\) are the usual choices for numbers. Nothing forces them. It is often clearer to pick a letter that means something: \(d(t)\) for distance as a function of time, \(W(t)\) for the temperature of water, \(V(t)\) for the number of visitors.

Three things in four symbols. The expression \(f(x)\) is carrying more than it looks like:

Keep those three straight and the rest of this subsection is bookkeeping. Confuse the first row with the third — treat the name of the rule as though it were a number — and you get the single most damaging error in the whole chapter, which is what comes next.

\(f(x)\) does not mean \(f\) times \(x\)

This misreading is close to universal, and it will wreck everything downstream, so kill it now.

The parentheses in \(f(x)\) indicate the input. They do not indicate multiplication.

You have spent years reading \(3(x)\) as "3 times \(x\)," so the instinct is honest — and wrong here. In \(f(x)\), the letter \(f\) is not a number that could be multiplied by anything. It is the name of a rule. The parentheses are doing the job that parentheses do in a calculator or a spreadsheet: they hold the thing being fed in.

Inputs do not have to be numbers

Function notation is more flexible than it first looks. Let \(f\) be the rule whose input is the name of a month and whose output is the number of days in that month, ignoring leap years. Then

because March has 31 days. The input is a word and the output is a number, which is no obstacle — the definition never asked for anything more.

The same goes for the class-schedule function: if \(f\) takes a time on Monday and returns the class you are in at that time, then a student who has English at 10:00 is described by

Not \(f(\text{English}) = 10{:}00\), which would be a different function altogether — one that takes a class name and returns a time, and which fails the test the moment a student has the same class twice in a day. Most functions in this course take numbers in and give numbers out, but the definition never required it.

Writing a relationship in function notation

To express "the weight of a pig, in pounds, is a function of its age in days," name the quantities and then name the function:

• let \(a\) be the age in days (input),
• let \(w\) be the weight in pounds (output),
• then \(w = f(a)\).

The pattern is always the same: output \(=\) function name (input). Getting it backwards is the single most common error on this skill, so check yourself with the question "which one am I choosing?" — that one goes inside the parentheses.

Four statements, written correctly. The last column shows what the reversed version would claim instead:

Every row of that table was decided the same way: find the quantity you are choosing, and put it inside the parentheses. Nothing about the letters themselves tells you which is which — \(m\), \(t\), \(T\), and \(h\) are just names — so the decision always comes back to the plain-English question from §1.3.1 about which quantity does the determining.

Evaluating a function

Naming a function is only useful if you can get a number out of it, which is the next move.

Definition 1.3.7 — Evaluating a function: replace every t with 12, and the rule hands back 48.

Marisol collected data on how far she walks each second and modelled her walking with the rule

where \(t\) is time in seconds and \(f(t)\) is distance in feet. To find how far she walks in 12 seconds, replace every \(t\) with 12:

She walks 48 feet in 12 seconds; the same substitution gives \(f(16) = 4(16) = 64\) feet. Notice what happened to the parentheses on the way through: on the left, \(f(12)\) means "feed 12 to \(f\)"; on the right, \(4(12)\) genuinely does mean "4 times 12." Both notations are on the same line and they mean different things. That is exactly why \(f(x)\) has to be read as a name-plus-input rather than a product.

A second example, and one you have already met. Jamal has been tracking the rodent population in the field behind the house, \(t\) weeks after spring begins, and it is behaving exactly like §1.2's Growing, Growing Dots, \(N(t) = 2^{t}\) — except Jamal's count starts at 8 rodents instead of 1, so every output of the dots pattern is scaled up by that same factor of 8:

That is worth pausing on. The doubling you built from a picture in §1.2 has not changed at all; it has only been given a name, a letter, and a way to be asked questions. Function notation is what lets you ask §1.2's pattern about week 16 without redrawing sixteen minutes of dots.

Evaluating the rodent model. Each row substitutes one value of \(t\) and finishes the arithmetic:

Those numbers are what the model says; whether that is what the field does is another question entirely. Predators, disease, food supply, and a hard winter would all pull the real count away from the model. A function is a rule, and a rule can be a good description of reality or a poor one.

Running the function backwards

Evaluating goes input \(\rightarrow\) output. The reverse question — output \(\rightarrow\) input — is asked with the whole function on the left and a number on the right.

reads "for what number of weeks is the population 128?" It is not an evaluation; it is an equation to solve. From the table above, \(t = 4\).

One character apart. The difference in notation is tiny and the difference in task is not:

Marisol's walking function shows the same pair. Writing \(f(16)\) asks how far she walks in 16 seconds. Writing \(f(t) = 200\) asks how long it takes her to walk 200 feet. Same function, opposite directions.

When the number on the right does not appear in a table, you solve. Jamal wants to know when the population passes 20,000, and neither \(p(11)\) nor \(p(12)\) is printed anywhere, so the two of them have to be computed and compared against the target. That kind of bracketing — find a week where the count is still too low, find a week where it is too high, and squeeze — is a perfectly respectable way to answer a question you cannot solve by algebra yet. It also keeps you honest about what the answer means: the population does not jump from 16,384 to 32,768 the instant week 12 begins, it climbs through 20,000 somewhere inside that week. Reporting "during week 12" says exactly what the two computed values support and no more.

1.3.6 Interpreting Function Notation in Context

Evaluating a function is arithmetic. Saying what the answer means is the actual skill, and it is the one the standards are most insistent about.

What \(f(3) = 7\) means

By itself, \(f(3) = 7\) says only this: when \(f\) takes 3 as its input, its output is 7. That is genuinely all. It is not much.

Attach quantities and units, and the same four symbols become a sentence about the world.

Suppose \(P\) gives the perimeter of a square whose side length is \(s\), with both measured in inches. Then

means "a square with side length 3 inches has a perimeter of 12 inches." And the backwards version, \(P(s) = 32\), means "a square with side length \(s\) inches has a perimeter of 32 inches" — from which you can reason that \(s\) must be 8, and then write \(P(8) = 32\).

Now change what the quantities are, keeping the numbers identical. Suppose \(f\) gives the number of blog subscribers, in thousands, \(m\) months after a blogger started publishing. Then

means "3 months after the blogger started, the blog has 12,000 subscribers."

Same symbols, wildly different sentence. The notation carries no meaning on its own; the meaning lives in what the input and output stand for, and somebody has to hand you that in the setup sentence. So read the setup twice before you touch the symbols.

Units will get you if you are careless

That blog function is measured in thousands. Miss that and every interpretation you write will be off by a factor of a thousand. Before interpreting any statement in function notation, answer three questions in order:

Question 3 matters more than it looks. If \(P(t)\) gives smartphone owners in millions, \(t\) years after 2000, then an input of \(17\) names the year 2017, an output of \(2320\) is 2.32 billion people rather than 2,320 people, and a negative input is not an error at all — it names a year before the counting started. The notation will not warn you when you get any of the three wrong.

A gallery of interpretations

Each row is one statement in function notation and the sentence it means. Read the middle column, cover the right column, and say the sentence yourself.

Every one of these rows is only readable because somebody told you what the input counts, what the output counts, and where the counting begins.

Comparing two outputs

Function notation can be put inside an inequality, and this is where interpretation gets genuinely useful.

Let \(W(t)\) give the temperature in degrees Fahrenheit of a pot of water on a stove, \(t\) minutes after the stove is turned on.

Read \(W(5) > W(2)\) carefully. It compares outputs, not inputs. It is not saying 5 is bigger than 2 — that was never in question. It is saying the temperature at minute 5 was higher than the temperature at minute 2, which is a real claim about the water and could have been false.

Reading it off a graph

Because every point on a graph has coordinates (input, output), function notation and graph reading are the same activity.

Let \(f(t)\) give the depth of water in a tub, in inches, \(t\) minutes after it started draining. Suppose the graph passes through \((0, 6)\), \((2, 5)\), and \((6, 3)\), and that it is level between \(t = 7\) and \(t = 10\).

Look hard at that last row. Two different inputs, \(7\) and \(10\), produce the same output. A whole flat stretch of graph is nothing but repeated outputs — and \(f\) is a function throughout, because no vertical line hits it twice. This is the misconception from §1.3.2, sitting in a real graph, where you can see it.

The same reading works on a table: every question you can ask about Table 1.3.1 — what \(g\) is at a given input, which inputs produce a particular output, which of the two functions is larger over a stretch — is answered in function notation, which is the whole point of having it.

Everything in this section has been about the rule: which inputs go with which outputs, and how to write that down — including, by the end, how to compare two rules' outputs with a single inequality like \(g(x) > f(x)\). §1.4 needs that move right away, the moment it puts two pools draining side by side on the same pair of axes. But first it asks the two questions that come next, and it can only ask them now that "function" means something exact.

The first is: which inputs are allowed? Sylvia's pool graph runs from the moment she started until the moment she finished, and no further. Jamal's population model takes weeks, and a negative number of weeks is meaningless in that story. A sequence takes term numbers, and there is no term 2.5. The collection of allowed inputs is called the domain.

The second is: which outputs actually occur? The water depth in Sylvia's pool never goes below zero and never goes above the rim. The letter-grade function only ever produces five letters. The collection of outputs that actually come out is called the range.

You have already used both words informally in this section, and you already have the tools to think about them, because both are questions about inputs and outputs — and you now know exactly what those are.

Problem Set 1.3

Key Terms