Integrated Math 1 · Chapter 1 · Functions

Inputs, Outputs, and What Makes a Function

One rule separates the relationships worth calculating with from the ones that are not: every input has exactly one output.

bookSHelf · Integrated Math 1 · §1.3 · a self-paced section

The definitional core of the chapter. Everything from §1.4 onwards assumes it.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

Outline — by the end of this section you will be able to

Objectives

Identify the input and the output, and say which quantity depends on the other Def 1.3.1–1.3.2
Read versus, with respect to, over, dependent on, and find the input Ex 1.3.1
State the definition of a function and apply it exactly Def 1.3.3
Explain why repeated outputs are allowed and one input with two is not Def 1.3.4
Decide from a table, ordered pairs, a mapping diagram, a graph, or words §1.3.3
Use the vertical line test on a graph, and explain why it works Def 1.3.5
Read, write and evaluate $f(x)$ , and interpret $f(3) = 7$ with units Def 1.3.6–1.3.7

The section SLOs, condensed from eight lines to seven.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.1 — Input and output

Two quantities, and one depends on the other

Time drives

The clock runs on its own. Sylvia's bucket does not speed it up or slow it down.

Depth follows

The water depth is entirely at the mercy of what Sylvia is doing, and when.

So the relationship has a direction, and that direction is the first thing to nail down about any pair of quantities — the test for a function, the graph, and the notation are all built on top of knowing which quantity does the determining.

Two quantities are never on equal footing. Direction first, everything else after.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

Insight — the knob and the dial

The push runs one way only

Picture an old radio. You turn the knob; the needle on the dial moves. Turning the knob is the input, and where the needle lands is the output.

Nobody grabs the needle and expects the knob to spin. That one-way push is exactly the relationship between time and water depth.

The image to keep for the whole section: knob first, needle second.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.1 — Definition

Input and Output

Definition 1.3.1 — Input and Output

In a relationship between two quantities, the input is the quantity you choose or are given, and the output is the quantity that is determined by it.

The output depends on the input; the input does not depend on the output.

Definition 1.3.1: you turn the knob and the needle follows, and the push runs one way only.

Time in, depth out. The reverse arrow is struck through in the figure on purpose.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.1 — Definition

Independent and Dependent Variables

Definition 1.3.2 — Independent and Dependent Variables

The independent variable is another name for the input — its value is not determined by the other quantity. The dependent variable is another name for the output — its value depends on the input.

Four words, two ideas. Teachers and textbooks switch between them without warning, so you need both.

Definition 1.3.2: four words, two ideas, and one question sorts every row.

If a question asks for the independent variable and you have been saying "input," you already have the answer.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.1 — One question sorts every row

Which quantity is which

Situation	Input (independent) — what you pick	Output (dependent) — what you get
Sylvia's pool	time since she started	depth of the water
Buying gas	number of gallons pumped	cost in dollars
A circle	length of the radius	area of the circle
A course grade	percent earned	letter grade
Floating down the river	time since the trip began	distance traveled

Table 1.3.1: Five of the six situations in §1.3.1, sorted the same way every time.

Ask 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.

The sixth row, Sylvia's pool, is already on the slide: time in, depth out. Floating down the river works the same way.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

Context Pause — why the wording is not optional

The words that tell you which is which

Phrasing	Output — named first	Input — after the phrase
"the cost of gas versus the amount pumped"	cost	gallons pumped
"temperature with respect to time of day"	temperature	time of day
"the national debt with respect to time"	national debt	time
"letter grade dependent on percent earned"	letter grade	percent earned

Table 1.3.2: The output is named first; the input is named after the connecting phrase.

Every graph you meet from here on has axes somebody chose, and the phrase in the problem is how you know which quantity they put on the horizontal axis. Read the phrase wrong and you plot the whole thing sideways, then answer a question nobody asked.

"Distance versus time," "with respect to," "over," "dependent on" — all four describe the same relationship.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.1 — Worked example

Pulling the input out of a phrase

Example 1.3.1 — name the input and the output in each phrase

a) The volume of water in a given cylinder is dependent on the height of the water.

b) The length of fence needed with respect to the rectangular area enclosed.

(a) The phrase is "dependent on," and height comes after it. Input: the height. Output: the volume. Pour water in until it reaches 10 inches and the volume is settled — you did not get to choose it separately.

(b) The phrase is "with respect to," and area comes after it. Input: the area enclosed. Output: the length of fence. This feels backwards, but the phrase settles it — and §1.3.3 comes back to this exact relationship.

Part (b) is planted deliberately — it is the counterexample in Definition 1.3.4.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.1 — Your turn

Try It Now 1.3.1

Try It Now 1.3.1 — name the input and the output

a) The cost of gas versus the amount of gas pumped. b) The area of a circle as it relates to the radius. c) A student's letter grade dependent on the percent earned.

d) Now reverse (c): the percent earned, dependent on the letter grade. What goes wrong?

(a) input gallons, output cost. (b) input radius, output area. (c) input percent, output letter grade.

(d) Feed in the letter A. Which percent should come back — 90, 94, or 100? All of them earned an A, so the reversed relationship has no single answer to give.

Part (d) is the whole of §1.3.2 in one question.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.2 — What makes a relationship a function

Reverse it and the behaviour changes

Name in, Social Security number out

Hand over "David Smith" and ask for the number. There are many people named David Smith, each with a different number. Several answers.

Number in, name out

Hand over a specific number and ask for the person. A Social Security number belongs to exactly one person. One answer.

Same two quantities, opposite behaviour. Everything in §1.3.1 was setup; the rule that makes that difference official is short enough to memorize in one reading.

Swapping input and output gives a different relationship, and the new one may behave completely differently.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.2 — Definition

Function

Definition 1.3.3 — Function

A function is a relationship between two quantities in which every input has exactly one output. If $x$ is an input of a function $f$ , then $f(x)$ is the one output that $f$ assigns to $x$ .

Every input — not most. Exactly one output — not "at least one," not "at most one."

Definition 1.3.3: two different inputs sharing an output is fine; one input with two different outputs is fatal.

The figure carries both halves on purpose. Read the two arrows arriving at 7 before the two arrows leaving one input.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.2 — The misconception that has to die first

Outputs may repeat. Inputs may not.

Two different inputs sharing an output — fine

Nothing in the definition forbids it. Ask about either input, get the same 5, write down 5. No confusion at all.

One input with two different outputs — fatal

That is exactly what the definition forbids. A 5 sometimes and a 9 other times, and the relationship has told you nothing.

Almost everyone first remembers the definition as "no repeats." That is half right, and the wrong half is the half people apply — the test only ever looks at the input column.

The asymmetry is there because a function has to answer a question with one answer you can write down.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

Insight — a vending machine, not a slot machine

Same button, same bag of chips

Press B4 on a vending machine and a specific bag of chips comes out, every time. That is a function. A slot machine takes the same pull and returns something different each time, which is exactly why no one would call it a function.

Two buttons dispensing the same chips is fine; one button dispensing something different each press is not.

Both halves again, in one image: two buttons, one snack — fine. One button, two outcomes — not.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.2 — The repeats are right in front of you

A repeated output, in a real table

$x$	$f(x)$	$g(x)$
$-1$	$6$	$3$
$0$	$2$	$7$
$1$	$0$	$11$
$2$	$0$	$15$
$3$	$2$	$19$
$4$	$6$	$23$

Table 1.3.3: Six rows of two continuous functions $f$ and $g$ at the same inputs.

Look at the $f(x)$ column: the output $0$ appears twice, at $x = 1$ and at $x = 2$ . The output $2$ repeats as well, and so does $6$ .

$f$ is still perfectly a function, because reading down the table no single $x$ value has two entries beside it.

The full table on the page runs from

x = -5

to

x = 6

; these six rows carry every repeat that matters.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.2 — Definition

Counterexample

Definition 1.3.4 — Counterexample

A counterexample is a single case that shows a general claim is false. To disprove "this relationship is a function," one counterexample is enough: one input with two different outputs.

You do not have to check the other thousand inputs once you have found the one that breaks it.

Definition 1.3.4: one input of 36 square feet, three different fence lengths, and the claim is dead.

A

6 \times 6

needs 24 feet, a

4 \times 9

needs 26, a

2 \times 18

needs 40 — same 36 square feet.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.2 — Two rules, side by side

Both tables look tidy. That is the trap.

Input (name)	Output (graduation year)
Taylor Swift	2008
Zendaya	2015
Storm Reid	2020
Johnny Orlando	2021

Table 1.3.4: Rule G — name in, graduation year out.

Input (graduation year)	Output (a person)
2020	Lil Tecca
2021	Olivia Rodrigo
2022	Forrest Wheeler
2023	Miles Brown

Table 1.3.5: Rule P — graduation year in, a person out.

A short list of pairs can always be written down neatly. What matters is what the rule does with an input that is not on the list — so run your own name and your own graduation year through both.

The test asks about every input, not about the four the author happened to print.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.2 — Worked example

Applying the test to Rule G and Rule P

Example 1.3.2 — decide whether Rule G is a function, and whether Rule P is

Rule G, with a general input. Feed in your own name. You graduated from high school once, so there is exactly one year to hand back. Every input gets exactly one output, so Rule G is a function.

Rule P, with a general input. Feed in your own graduation year. Millions of people graduated that year, and there is no single right answer to hand back.

Name the counterexample. Take the input 2020. Rule P could return Lil Tecca, or any of the millions of other 2020 graduates. One input, more than one output — so Rule P is not a function.

Answer: Rule G is a function; Rule P is not, and 2020 is the counterexample.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

Context Pause — why schools use ID numbers

One honest caution about Rule G

Two different people can be named David Smith. So if "name" means "a string of letters," Rule G has Rule P's problem: one input, more than one graduation year.

The rule is a function only when the input identifies exactly one person — which is precisely why a district looks up a transcript by student ID and not by name.

A real design decision that falls straight out of the definition.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.2 — Your turn

Try It Now 1.3.2

Try It Now 1.3.2 — decide, and explain

a) Input a US state, output its capital city. b) Reverse it: input a city, output a state that city is in.

c) In the table of $f$ and $g$ , the output $6$ appears twice in the $f(x)$ column. Does that make $f$ fail the test?

(a) Yes. Each state has been assigned exactly one capital by law — one output, never two.

(b) No. Springfield is a city in Illinois, and in Missouri, and in Massachusetts, and in about thirty other states. Springfield is the counterexample.

(c) No. Two different inputs landing on the same output is exactly what the repeat rule allows. No $x$ in that table has two values beside it.

Part (c) is the misconception, asked directly.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.3 — Testing every representation

Five costumes, one test

A relationship can arrive as a set of ordered pairs, a table, a mapping diagram, a graph, or a description in words. The test never changes; only what you are looking at does.

Ordered pairs — in $(x, y)$ the first number is the input

$\{(-7, 2),\; (3, 5),\; (8, 4),\; (-6, 5),\; (-2, 3)\}$

First numbers $-7, 3, 8, -6, -2$ — all different, so this is a function. The output $5$ appears twice, from $3$ and from $-6$ , which is allowed and irrelevant.

$\{(9, 2),\; (0, 4),\; (4, 0),\; (5, 3),\; (2, 7),\; (0, -3),\; (3, -1)\}$

The input $0$ appears twice, paired with $4$ and with $-3$ . Not a function — $(0, 4)$ and $(0, -3)$ are the counterexample.

Does any first number appear twice with two different second numbers? That is the whole test.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

Insight — one test, five disguises

The fingerprint does not change with the surface

Think of a fingerprint check. The fingerprint is the same whether you lift it off a glass, a doorknob, or a phone screen — only the surface changes.

"One input, two outputs" is the fingerprint. A table, a list of pairs, a diagram, a graph, and a sentence are just five surfaces to look for it on.

Learn the fatal pattern in each costume and you can decide any of them at a glance.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.3 — A table is a set of ordered pairs written vertically

Read down the input column, never across

$x$	$h(x)$
$1$	$9$
$2$	$98$
$3$	$987$
$4$	$9876$

Table 1.3.6: Inputs $1, 2, 3, 4$ , all different. A function.

$x$	$f(x)$
$0$	$2$
$1$	$3$
$2$	$0$
$3$	$2$
$4$	$6$

Table 1.3.7: Each input once. A function — and $2$ repeats.

The question is never "does the output column repeat." It is only ever "does one input have two outputs."

The full

f

table on the page runs to

x = 6

; rows 5 and 6 add nothing new.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.3 — A mapping diagram makes the test visual

Arrows arriving, versus arrows leaving

Two arrows arrive at 7 — a function

$1 \rightarrow 7$ , $2 \rightarrow 9$ , $3 \rightarrow 7$ , $4 \rightarrow 12$ . Every input has exactly one arrow leaving it. Arrows landing in the same place is the repeated-output situation, which is fine.

Two arrows leave 2 — not a function

$1 \rightarrow 7$ , $2 \rightarrow 9$ , $2 \rightarrow 5$ , $4 \rightarrow 12$ . The input $2$ is headed to $9$ and to $5$ at once. Arrows leaving the same place is the fatal one.

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

Only one line changed between the two lists — that is the whole difference.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

Context Pause — one test you can run with a ruler

The only costume you can check without reading a number

Lay a ruler vertically on the graph, slide it left to right, and watch how many times it touches.

That is the entire test, and it is why graphs are the fastest way to decide. Every point on a graph is a pair (input, output): the $x$ -coordinate is the input, the $y$ -coordinate is the output.

Two points with the same

x

and different

y

sit on the same vertical line — that is one input with two outputs, drawn.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.3 — Definition

Vertical Line Test

Definition 1.3.5 — Vertical Line Test

A graph represents a function if and only if no vertical line crosses the graph more than once.

Why it works is not a separate fact. A vertical line is the set of all points sharing one $x$ -value, so a vertical line is a single input, and each crossing is an output assigned to it. The test is the definition, drawn.

Definition 1.3.5: a vertical line is one input, so two crossings means that input has two outputs.

Not a rule to remember alongside the definition — the same rule, in a picture.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.3 — Two consequences that look symmetric and are not

A horizontal line passes. A vertical line fails.

$y = 8$ is a function

Every vertical line crosses it exactly once, handing back the output $8$ for every input. Repeated outputs, in the extreme case — and still perfectly legal.

$x = 4$ is not

The vertical line at $x = 4$ lies on top of it and crosses everywhere at once, so the input $4$ is answering "every number at the same time."

A circle is the same failure in a friendlier shape. On the circle of radius 5 at the origin, the vertical line $x = 3$ meets it at $(3, -4)$ and $(3, 4)$ , since $3^2 + 4^2 = 25 = 5^2$ . One input, two outputs.

If you can explain that asymmetry, you understand the definition.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.3 — Your turn

Try It Now 1.3.3

Try It Now 1.3.3 — decide, and name any counterexample

a) Is $\{(2, 5),\; (4, 5),\; (6, 5),\; (8, 5)\}$ a function? b) Is $\{(4, 1),\; (5, 2),\; (4, 3),\; (6, 7)\}$ ?

c) A graph is a straight line slanting up from lower left to upper right. Does it pass the vertical line test?

(a) Function. Inputs $2, 4, 6, 8$ are all different. The output $5$ repeats four times, which the repeat rule allows.

(b) Not a function. The input $4$ is paired with $1$ and with $3$ ; the counterexample is $(4, 1)$ together with $(4, 3)$ .

(c) Yes. Wherever you put the vertical line it meets the graph exactly once — a slanted line never doubles back over the same $x$ .

Part (d) on the page asks about °F to °C: one temperature is never two temperatures at once, so it is a function.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.4 — A catalogue of relationships

Run the definition across a pile of situations

Relationship	Input → Output	Function?	Why
A Social Security number, and the person	SSN → name	yes	a number belongs to exactly one person
A person's name, and their SSN	name → SSN	no	several people share a name
The area of a circle, from the radius	radius → area	yes	$A = \pi r^2$ returns one area
Length of fence, from area enclosed	area → length	no	many rectangles share an area
Stadium a player is in, and the outcome	stadium → outcome	no	many games, different outcomes
Height on a Ferris wheel, over time	time → height	yes	at one instant, one height

Table 1.3.8: Six of the sixteen relationships tested in §1.3.4. Name the two quantities, decide which is the input, then decide.

The full sixteen are on the page; these six carry every distinct reason a relationship passes or fails.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

Insight — collapsing is a one-way door

A shredder runs in one direction

Many different sheets go in and the same pile of confetti comes out — that direction works fine. Ask for the confetti to become one specific sheet again and there is no answer.

A relationship that collapses inputs together is a shredder, and its reverse is never a function. That is exactly why "name → SSN" fails while "SSN → name" passes.

Collapsing many inputs onto one output is legal; undoing it is not.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.4 — Your turn, and one stretch: a sequence is a function

Try It Now 1.3.4

Try It Now 1.3.4 — movies, and a sequence

a) Input a movie title, output the year it was released. b) Input a year, output a movie released that year.

c) A sequence begins $2, 9, 16, 23, \dots$ , adding 7 each time. Write a formula for the $n$ th term and find the 10th.

(a) Function. A given movie came out in one year. Two movies can share a release year — a repeated output, therefore fine.

(b) Not a function, and 2019 is the counterexample: hundreds of movies came out that year. The shredder again.

$f(n) = 2 + 7(n - 1) = 7n - 5 \qquad f(10) = 70 - 5 = 65$

A sequence is a function whose inputs can only be whole numbers — §1.4 will call that its domain.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.5 — Function notation

Give the function a letter, then attach the input

Writing "the depth of the water in Sylvia's pool 40 minutes after she started" every time is not sustainable. Once you know a relationship is a function, you need a compact way to talk about it — and the pattern never changes:

$\text{output} = \text{function name}(\text{input})$

Pick a letter that means something

$f$ , $g$ , $h$ are the usual choices, but $d(t)$ for distance over time and $V(t)$ for visitors are clearer.

Inputs do not have to be numbers

If $f$ takes a month and returns its length in days, then $f(\text{March}) = 31$ . The definition never asked for more.

A class-schedule function:

f(10{:}00) = \text{English}

, not the reverse.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.5 — Definition

Function Notation

Definition 1.3.6 — Function Notation

The notation $y = f(x)$ defines a function named $f$ . It is read " $y$ equals $f$ of $x$ ." The letter $x$ inside the parentheses is the input, or independent variable; $y$ , also written $f(x)$ , is the output, or dependent variable.

Three things in four symbols: $f$ is the rule, $x$ is the input, $f(x)$ is the output.

Definition 1.3.6: $f$ names the rule, $x$ names the input, and $f(x)$ names the output.

Confuse the rule with the output and you get the single most damaging error in the chapter — which is the next slide.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

Insight — a label on a box, not a factor

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

Think of $f$ as the label printed on a machine and $x$ as what you drop in the hopper. Nobody multiplies a blender by a banana. You put the banana in the blender and something comes out, and $f(x)$ is the name of what comes out.

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

This misreading is close to universal and it wrecks everything downstream. Kill it here.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.5 — Worked example

Writing a relationship in function notation

Example 1.3.3 — name your variables first, then write it

a) The tons of garbage $G$ produced in a week by a city of population $p$ .

b) The cubic yards of dirt $D$ needed to cover a garden of area $a$ square feet.

(a) You pick a city, and its population comes with it; the garbage total is then whatever it turns out to be. Input $p$ , output $G$ . (b) You measure the garden first, then figure out how much dirt to order. Input $a$ , output $D$ .

$G = f(p) \qquad D = g(a)$

p = f(G)

would claim that weighing the garbage tells the city how many people to have.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.5 — Definition

Evaluating a Function

Definition 1.3.7 — Evaluating a Function

To evaluate a function at an input means to substitute that input into the rule and compute the output.

Marisol walks $f(t) = 4t$ feet in $t$ seconds. To find 12 seconds, replace every $t$ with 12:

$f(12) = 4(12) = 48$

Definition 1.3.7: replace every $t$ with 12, and the rule hands back 48.

On that one line,

f(12)

means "feed 12 to

f

" and

4(12)

genuinely does mean "4 times 12."

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.5 — A second rule, this one exponential

Evaluating $p(t) = 8 \cdot 2^{\,t}$

Question	Substitution	Result	In words
$p(4)$	$8 \cdot 2^4 = 8 \cdot 16$	$128$	128 rodents after 4 weeks
$p(10)$	$8 \cdot 2^{10} = 8 \cdot 1024$	$8192$	8,192 rodents after 10 weeks
$p(16)$	$8 \cdot 2^{16} = 8 \cdot 65{,}536$	$524{,}288$	524,288 after a 16-week summer

Table 1.3.9: 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. A function is a rule, and a rule can be a good description of reality or a poor one.

Predators, disease, food supply and a hard winter all pull the real count away from the model.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

Context Pause — which side the letter is on

One character apart, two different tasks

Written	Asks	Type of question
$p(4)$	what is the population at week 4?	evaluate — substitute and compute
$p(t) = 128$	at what week is the population 128?	solve — find the input

Table 1.3.10: Evaluating goes input → output. The reverse question puts the whole function on the left and a number on the right.

A number inside the parentheses means you are being handed an input and asked to compute. A letter inside, with a number after the equals sign, means you are being handed an output and asked to hunt for the input. Read the parentheses before you start working.

f(16)

asks how far Marisol walks in 16 seconds;

f(t) = 200

asks how long 200 feet takes.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.5 — Worked example

Solving instead of evaluating

Example 1.3.4 — using $p(t) = 8 \cdot 2^{\,t}$ , find the first whole week the population exceeds 20,000

The number 20,000 is an output and the unknown is the input, so this is a solve. Try week 11, then week 12:

$p(11) = 8 \cdot 2048 = 16{,}384 \qquad p(12) = 8 \cdot 4096 = 32{,}768$

Under at week 11, over at week 12 — the population first exceeds 20,000 during week 12.

Bracketing — too low, too high, squeeze — is a respectable way to answer a question you cannot solve by algebra yet.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.5 — Your turn

Try It Now 1.3.5

Try It Now 1.3.5 — $d(t) = 78t$ gives the miles a family drives in $t$ hours

a) Find $d(4)$ and say what it means. b) Why is $f(x)$ not $f$ times $x$ ?

c) Write "the height $h$ of the water in a pool, in inches, is a function of the minutes $m$ since draining began." d) Which of $d(3.5)$ and $d(t) = 450$ is an evaluate, and which a solve?

$d(4) = 78(4) = 312$

(a) The family travels 312 miles in 4 hours. (b) $f$ is the name of a rule, not a number, so there is nothing there to multiply by. (c) You pick a minute after draining begins and the depth follows, so $h = f(m)$ is the notation.

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s

s

is the input.

(d) $d(3.5)$ hands you an input — evaluate. $d(t) = 450$ hands you an output and hides the input — solve.

Answers: 312 miles; a rule is not a factor;

h = f(m)

; evaluate then solve.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.6 — Interpreting function notation in context

What $f(3) = 7$ actually means

By itself it says only this: when $f$ takes 3 as its input, its output is 7. That is genuinely all. Attach quantities and units and the same four symbols become a sentence about the world.

Three questions before you interpret anything

1. What quantity is the input, and in what units?

2. What quantity is the output, and in what units?

3. Where does the input start counting from?

If $P(t)$ gives smartphone owners in millions, $t$ years after 2000, then an input of $17$ names 2017, an output of $2320$ is 2.32 billion people, and a negative input is not an error — it names a year before the counting started.

The notation carries no meaning on its own; somebody has to hand you the setup sentence.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.6 — Worked example

The same numbers, two different sentences

Example 1.3.5 — interpret each statement in words

a) $P(3) = 12$ , where $P(s)$ is the perimeter in inches of a square with side length $s$ inches.

b) $f(3) = 12$ , where $f(m)$ is blog subscribers, in thousands, $m$ months after publishing began.

(a) A square with side length 3 inches has a perimeter of 12 inches. Sanity-check it: four sides of 3 inches gives $4(3) = 12$ .

(b) Three months after the blogger started, the blog had 12,000 subscribers. That "in thousands" is the whole difference between a right answer and one off by a factor of a thousand — the output 12 is not 12 subscribers.

Same symbols, wildly different sentence. The meaning lives in what the input and output stand for.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.6 — Worked example

Getting the units and the starting point right

Example 1.3.6 — $P(t)$ gives smartphone owners worldwide, in millions, $t$ years after 2000

a) Interpret $P(17) = 2320$ . b) Write "in 2010, 296,600,000 people owned a smartphone." c) Interpret $P(-10) = 0$ .

(a) $t = 17$ is the year 2017, and 2320 million is 2.32 billion: in 2017, about 2.32 billion people owned a smartphone.

(b) 2010 is $10$ years after 2000, and 296,600,000 is 296.6 million, so $P(10) = 296.6$ .

(c) $t = -10$ is 1990, and an output of $0$ million is zero people: in 1990, no one owned a smartphone. The negative sign names a year before the counting started.

The input is 10, not 2010; the output is 296.6, not 296,600,000.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.6 — Worked example

Reading a table in function notation

Example 1.3.7 — using the table of $f$ and $g$ , find $g(-3)$ , and every input where $f(x) = 0$

Read down the $x$ column to the row $-3$ , then across to $g(x)$ . The second question runs backwards: scan the $f(x)$ column for zeros, then read back to the inputs that produced them.

$g(-3) = -5 \qquad f(1) = 0 \quad \text{and} \quad f(2) = 0$

Going input → output always gives one answer. Going output → input may give several — the repeated-output rule showing up as a practical difference, not a defect. A flat stretch of graph is the same thing: $f(7) = f(10)$ , and $f$ is a function throughout.

The misconception from §1.3.2, sitting in a real graph, where you can see it.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3.6 — Your turn

Try It Now 1.3.6

Try It Now 1.3.6 — $V(t)$ gives museum visitors $t$ hours after it opens at 9 a.m.

a) Interpret $V(4) = 257$ . b) Interpret $V(8) = 0$ . c) Write "at 10:15 a.m. there were 28 visitors."

d) $T(h)$ is the temperature of a house, $h$ hours after midnight. Interpret $T(6) > T(2)$ .

(a) At 1 p.m. there were 257 visitors. (b) At 5 p.m. there were none left, which is why 5 p.m. is closing time.

(c) 10:15 a.m. is $1.25$ hours after opening, so $V(1.25) = 28$ . (d) The house was warmer at 6 a.m. than at 2 a.m. — it compares outputs, not inputs.

Part (d) was never a claim that 6 is bigger than 2 — it is a claim about the temperature that could have turned out false.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

Key Terminology — the nine words this section defines

Key terms

input — the quantity you choose or are given.

output — the quantity determined by the input.

independent variable — another name for the input; its value is not determined by the other quantity.

dependent variable — another name for the output; its value depends on the input.

function — a relationship in which every input has exactly one output.

counterexample — a single case that disproves a general claim; here, one input with two different outputs.

vertical line test — a graph is a function exactly when no vertical line crosses it more than once.

function notation — $y = f(x)$ , naming the rule $f$ , its input $x$ , and its output $f(x)$ .

evaluating a function — substituting an input into the rule and computing the output.

Second column reveals on click.

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

The headline result

Every input has exactly one output

Two different inputs sharing an output is fine. One input with two different outputs is fatal. The test only ever looks at the input column, and one counterexample settles it.

That one sentence is what the vertical line test draws, what the mapping diagram shows, and what function notation assumes when it promises that $f(x)$ names the output.

† The half everybody drops is the first one. A repeated output creates no ambiguity — ask about either input, get the same answer, write it down.

Say both halves out loud. Showing only the forbidden case teaches the misconception.

1.3

Inputs, Outputs, and What Makes a Function · bookSHelf Integrated Math 1§1.3

§1.3 — Conclusions

What to carry forward

The one idea

One rule, five costumes. On a graph it is drawn as the vertical line test — a vertical line is one input, so two crossings means two outputs. Function notation just names the pieces: $f$ the rule, $x$ the input, $f(x)$ the output.

Where it goes wrong

Remembering the rule as "no repeats" and checking the output column; reading a table sideways; reversing input and output in the parentheses; reading $f(x)$ as a product; and interpreting a value without its units or its starting point.

Next: §1.4 asks which inputs are allowed (the domain) and which outputs actually occur (the range) — questions it can only ask now that "function" means something exact. Back to start.

Closes the definitional core of Chapter 1.

Inputs, Outputs, and What Makes a Function

Objectives

Two quantities, and one depends on the other

Time drives

Depth follows

The push runs one way only

Input and Output

Independent and Dependent Variables

Which quantity is which

The words that tell you which is which

Pulling the input out of a phrase

Try It Now 1.3.1

Reverse it and the behaviour changes

Name in, Social Security number out

Number in, name out

Function

Outputs may repeat. Inputs may not.

Two different inputs sharing an output — fine

One input with two different outputs — fatal

Same button, same bag of chips

A repeated output, in a real table

Counterexample

Both tables look tidy. That is the trap.

Applying the test to Rule G and Rule P

One honest caution about Rule G

Try It Now 1.3.2

Five costumes, one test

The fingerprint does not change with the surface

Read down the input column, never across

Arrows arriving, versus arrows leaving

Two arrows arrive at 7 — a function

Two arrows leave 2 — not a function

The only costume you can check without reading a number

Vertical Line Test

A horizontal line passes. A vertical line fails.

y=8y = 8y=8 is a function

x=4x = 4x=4 is not

Try It Now 1.3.3

Run the definition across a pile of situations

A shredder runs in one direction

Try It Now 1.3.4

Give the function a letter, then attach the input

Pick a letter that means something

Inputs do not have to be numbers

Function Notation

f(x)f(x)f(x) does not mean fff times xxx

Writing a relationship in function notation

Evaluating a Function

Evaluating p(t)=8⋅2 tp(t) = 8 \cdot 2^{\,t}p(t)=8⋅2t

One character apart, two different tasks

Solving instead of evaluating

Try It Now 1.3.5

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

The same numbers, two different sentences

Getting the units and the starting point right

Reading a table in function notation

Try It Now 1.3.6

Key terms

What to carry forward

The one idea

Where it goes wrong

$y = 8$ is a function

$x = 4$ is not

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

Evaluating $p(t) = 8 \cdot 2^{\,t}$

What $f(3) = 7$ actually means