Integrated Math 1 · Chapter 1 · Functions

Domain and Range

A rule will accept almost any number you hand it. The situation it models will not. This section is about the gap between those two answers — and about writing a discrete domain as the list it really is.

bookSHelf · Integrated Math 1 · §1.4 · a self-paced section

Domain and range are new names for §1.3's inputs and outputs — plus a sharper question.

Domain and Range · bookSHelf Integrated Math 1§1.4

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

Objectives

Name the domain as the set of allowed inputs and the range as the resulting outputs Def 1.4.1–2
Find a domain and range from a table, a list of ordered pairs, or a graph §1.4.3–4
Decide whether a number is a possible input or output, and say why Ex 1.4.1
Determine a reasonable domain, apart from what the rule alone would allow Def 1.4.3
Tell a discrete function from a continuous one, and write each domain to match Def 1.4.4–5
Identify intercepts, maximum and minimum, and say what each means in context Def 1.4.6–9

Six SLOs, one line each.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.1 — The Water Park

Two pumps, two pools, one pair of axes

Aly and Dayne each drain a pool at their water park, one pump apiece. One graph shows both at once: the water in Aly’s pool, $a(x)$ , and in Dayne’s, $d(x)$ , against the time $x$ in minutes since they started.

Before any equations — say what you see

Both pools are draining, each at a constant rate — straight segments. The rates are not the same: Aly’s is steeper, so Aly’s pool drains faster.

Dayne’s pool starts with less water, but the segments cross: at one moment both hold the same amount, and after it the order flips.

Both eventually reach zero. Both pools do get empty; they just do not get there at the same time.

Figure 1.4.1: two pools draining on one pair of axes — the steeper line starts higher and still empties first.

Every observation above came from the shape of two lines, with no scale on either axis. The section attaches the numbers on the next slides.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.1 — Worked example

Dayne's pump: building a rule from a rate

Dayne's pump removes 1000 gallons a minute and empties the pool in 24 minutes. Write $d(x)$ , the gallons left after $x$ minutes, and check it at both ends.

Step	Work
Start amount	At 1000 gallons a minute for 24 minutes, the pool started with $1000 \cdot 24 = 24{,}000$ gallons.
Write the rule	After $x$ minutes the pump has removed $1000x$ gallons, so $d(x) = 24{,}000 - 1000x$ .
Say the pieces	$24{,}000$ = water at the start; $1000$ = the rate; $-1000x$ = the total removed — the minus sign is the draining.
Check both ends	$d(0) = 24{,}000$ and $d(24) = 24{,}000 - 1000(24) = 0$ . At the start the pool is full; after 24 minutes it is empty.

Answer. $d(x) = 24{,}000 - 1000x$ , and both endpoints agree with the story, so the rule is right.

Example 1.4.1. Two facts about a rate are enough to write the function.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.1 — Worked example

Aly's pump: building a rule from two points on a graph

Aly's pump does not come with a stated rate. It comes with a graph: Aly's segment starts at 28,000 gallons when $x = 0$ and reaches zero at 20 minutes. Write $a(x)$ .

Step	Work
Find the rate	Two points are enough for a straight line, and the two easiest to read are the ends. Aly's pump removed 28,000 gallons in 20 minutes: $\frac{28{,}000 \text{ gallons}}{20 \text{ minutes}} = 1400$ gallons per minute.
Same shape as Dayne's	Start full, subtract the rate times the time: $a(x) = 28{,}000 - 1400x$ .
Check the ends	$a(0) = 28{,}000$ and $a(20) = 28{,}000 - 1400(20) = 0$ .

Answer. $a(x) = 28{,}000 - 1400x$ . Aly's pool started with 4000 more gallons and drained 400 gallons per minute faster, which is why it finished four minutes sooner.

Example 1.4.2. Reading a rule off a graph is the skill §1.2 has been building. Keep both rules where you can see them.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.1 — Your turn

Try It Now 1.4.1

Try It Now 1.4.1

A third pump empties a 30,000-gallon pool at 1500 gallons a minute.

a) Write the rule for the gallons remaining after $x$ minutes.

b) How long does it take to empty?

c) How much is left after 12 minutes?

Answer

a) Start full, subtract the rate times the time: $p(x) = 30{,}000 - 1500x$ .

b) The pool is empty when $p(x) = 0$ : $30{,}000 - 1500x = 0$ , so $1500x = 30{,}000$ and $x = 20$ minutes.

c) $p(12) = 30{,}000 - 1500(12) = 30{,}000 - 18{,}000 = 12{,}000$ gallons.

Try It Now 1.4.1. The third pump returns in Try It Now 1.4.2.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.2 — Domain: the set of allowed inputs

Not “all numbers” — all the possible ones

Which inputs are possible has to be answered fresh for every function, by looking at both the rule and the situation. Three shapes cover almost every domain you will write.

Shape of the domain	Example	How to write it
A short list of specific values	camp enrollment	$\{5, 6, 7, \ldots, 16\}$
Everything from one number on	side length of a square	$s \ge 0$
Everything except a few values	$f(x) = \dfrac{6}{x-2}$	all real numbers, $x \neq 2$

A sentence in plain English is always allowed too: “any whole number of students from 5 to 16” says exactly what $\{5, 6, \ldots, 16\}$ says.

Three shapes, three notations. The list form is the one students smear into an interval.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.2 — Definition

Domain

Definition 1.4.1 — Domain

The domain of a function is the set of all of its possible input values.

Notice what it does not say. It does not say “all numbers.” It says all the possible inputs, which leaves open the question of what makes an input possible.

Definition 1.4.1: sweep left to right, and the band you lay down along the input axis is the set of allowed inputs.

The sweep is the technique §1.4.4 formalises; here it just makes the definition visible.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.2 — Your turn

Try It Now 1.4.1

Try It Now 1.4.1

A movie theater sells tickets for a single showing; the auditorium holds 240 people.

a) Describe the domain of the ticket-sales function in words.

b) Find the domain of $g(x) = \dfrac{10}{x + 4}$ .

Answer

a) People come in whole units, you cannot sell a negative ticket, and the room holds 240: the whole numbers $\{0, 1, 2, \ldots, 240\}$ . Both limits came from the situation.

b) No situation here, so only the arithmetic can refuse. $x + 4 = 0$ at $x = -4$ , so all real numbers with $x \neq -4$ .

Try It Now 1.4.1. One domain from a situation, one from the rule.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.3 — Range: the set of resulting outputs

The range follows from the domain

If you have the domain and the rule, the range is not a new mystery — it is what you get by running every allowed input through the machine and collecting the results.

For Dayne's pool the output is an amount of water in gallons. The most it ever holds is 24,000, at the start. The least is 0, at the end. It falls steadily between those two without skipping anything, so every amount in between actually occurs at some moment.

0 \le d \le 24{,}000

Aly's range is $0 \le a \le 28{,}000$ by the same argument. Domain and range look alike for these two pools. That is a coincidence of this function, not a rule — the next example is where the two come apart.

Domain in, range out. Same-looking answer here is a coincidence.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.3 — Definition

Range

Definition 1.4.2 — Range

The range of a function is the set of all of its possible output values.

Domain is what goes in; range is what comes out. The second definition is the mirror of the first, and so is the sweep that finds it.

Definition 1.4.2: sweep bottom to top, and the band you lay down along the output axis is the set of resulting outputs.

Mirror of Definition 1.4.1 — same graph, the other axis.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.3 — Worked example

Two range questions, one method

Is 2000 gallons a possible output for Dayne's pool? Is 25,000? To test a number, work backwards and ask which input produces it. Solve $d(x) = 24{,}000 - 1000x$ for each; what decides it is whether the input handed back is in the domain $[0, 24]$ .

Question	Work backwards	Verdict
Is 2000 in the range?	$24{,}000 - 1000x = 2000 \Rightarrow 1000x = 22{,}000 \Rightarrow x = 22$	yes — $x = 22$ is in $[0, 24]$ . The pool holds 2000 gallons at exactly 22 minutes, two minutes before it runs dry.
Is 25,000 in the range?	$24{,}000 - 1000x = 25{,}000 \Rightarrow 1000x = -1000 \Rightarrow x = -1$	no — that input is one minute before the pump switched on, and it is not in the domain. The pool never held more water than it started with.

Answer. 2000 is in the range (at $x = 22$ ); 25,000 is not. The arithmetic answered happily both times; the domain decided it.

Example 1.4.3. Solve, then ask whether the input you got is allowed. The domain decides.

Domain and Range · bookSHelf Integrated Math 1§1.4

Insight — reading the machine backwards

Ranges that surprise you

Testing a range value is like checking whether a vending machine can hand you a specific snack. You do not shake the machine — you look at which button would have to be pressed. If there is no such button, the snack is not on offer.

Function	Domain	Range
$p(x) = 3$ , for $x \ge -2$	infinitely many inputs	$\{3\}$
$f(x) = 2x - 3$	all real numbers	all real numbers
$g(x) = \dfrac{1}{x - 2}$	all real numbers, $x \neq 2$	all real numbers, $g \neq 0$

In the last row the excluded input and the excluded output are different numbers. A fraction with 1 on top is never zero.

An enormous domain can have a one-element range.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.3 — Your turn

Try It Now 1.4.2

Try It Now 1.4.2

A club sells wristbands for $3 each and has 20 to sell, so $W(b) = 3b$ .

a) State the domain and the range.

b) Is $46 a possible total? Use the backwards move.

Answer

a) Domain $\{0, 1, 2, \ldots, 20\}$ ; range $\{0, 3, 6, \ldots, 60\}$ — the twenty-one multiples of 3 up to $3 \cdot 20 = 60$ .

b) $3b = 46$ gives $b \approx 15.3$ , not a whole number of wristbands. The club can collect $45 or $48, never exactly $46.

Try It Now 1.4.2. Working backwards is the range test.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.4 — Writing a domain down: interval notation

A shorthand for “every time from here to there”

Inequalities like $0 \le x \le 24$ have a shorthand, and a draining pool produces exactly the domain it was made for. Dayne’s, both ends in, is $[0, 24]$ , and his range is $[0,\ 24{,}000]$ . Aly's are $[0, 20]$ and $[0,\ 28{,}000]$ .

Written	Says	In words
$[0, 24]$	$0 \le x \le 24$	from 0 to 24, both endpoints included
$(0, 24)$	$0 < x < 24$	strictly between 0 and 24, neither endpoint included
$[0, 24)$	$0 \le x < 24$	0 is included, 24 is not
$[0, \infty)$	$x \ge 0$	0 and everything above it, forever

The symbol $\infty$ never gets a square bracket: that would say “this endpoint is in the set,” and infinity is a direction, not a number. Which bracket is a question about the situation: at $x = 0$ the pump switches on and the pool holds 24,000 gallons, a real reading, so the bracket is square.

The bracket carries the whole meaning. Four shapes, one table.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.4 — Definition

Interval Notation

Definition 1.4.3 — Interval Notation

Interval notation writes a set of numbers by giving its two endpoints, with a square bracket where the endpoint is included and a round bracket where it is not.

The bracket carries the whole meaning. Is the pool's starting moment part of the story? Yes — at $x = 0$ the pump switches on and the pool holds 24,000 gallons, which is a real reading. So the bracket is square. Is the finishing moment part of the story? Also yes: at $x = 24$ the pool holds 0 gallons, which is a real reading too.

Definition 1.4.3: filled means included, open means excluded — and the bracket says which.

Definition 1.4.3. Bracket-versus-parenthesis is the payload; every later domain in the deck is written with it.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.4 — Your turn

Try It Now 1.4.4

Try It Now 1.4.4

Write each in interval notation, or explain why you cannot.

a) $-2 \le x \le 7$

b) $x > 0$

c) all real numbers except 4

Answer

a) Both endpoints are included, so both brackets are square: $[-2, 7]$ .

b) Zero itself is excluded and there is no upper end, so a round bracket on both sides: $(0, \infty)$ . Infinity always takes a round bracket.

c) You cannot write this as a single interval. An interval is one unbroken stretch, and this set is the number line with a hole punched in it at 4. It needs two intervals joined together, or the plain description “all real numbers except 4.”

Try It Now 1.4.4. Part c is the handoff to §1.4.7 — a gap is not an interval.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.5 — Reasonable domain and range in context

Two domains compete for the name

Question	Answer for $T(n) = 7n$ , $7 tickets, 100 seats
What would the rule accept?	Any number at all. $7 \cdot (-3) = -21$ ; $7 \cdot 2.5 = 17.5$ . The arithmetic never complains.
What does the situation permit?	Whole numbers only, at least 0, at most 100.

The second is almost always smaller, and the second is the one you want. It is the answer the model is actually making a claim about.

The whole subsection in two rows.

Domain and Range · bookSHelf Integrated Math 1§1.4

Context Pause — this is a judgment, not a calculation

Same thirty seconds, four different ranges

A child swings for 30 seconds. Every function below has the same input — time since the child got on — so every domain is the same. Only the choice of output quantity differs, and no formula produces these answers.

	Output quantity	Domain	Range
$h$	height of the seat above the ground, ft	$0 \le t \le 30$	$1.5 \le h \le 4$
$r$	time left on the swing, s	$0 \le t \le 30$	$0 \le r \le 30$
$d$	distance from the top beam, ft	$0 \le t \le 30$	$\{7\}$
$p$	total number of pushes so far	$0 \le t \le 30$	$\{0, 1, 2, 3, 4, 5\}$

Reading the situation is the skill being tested. Note p jumps; there is no such thing as 2.5 pushes.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.5 — Definition

Reasonable Domain

Definition 1.4.3 — Reasonable Domain

The reasonable domain of a function is the set of input values that make sense in the situation being modeled, which may be much smaller than the set the rule alone would accept.

Time cannot be negative. You cannot buy $2.5$ buses. A rocket’s height stops meaning anything once it lands.

Definition 1.4.3: the rule accepts an unbroken number line, and the situation clips it three times.

Three clips: whole units, a floor, and a ceiling.

Domain and Range · bookSHelf Integrated Math 1§1.4

Insight — the rule is the door, the situation is the bouncer

Three functions, three different reasons

The rule only refuses numbers that break the arithmetic. The situation stands outside and turns away numbers the arithmetic would have accepted without complaint.

Function	Domain	Who refused
$A(s) = s^2$ , area of a square	$s \ge 0$	the situation — $s^2$ squares $-4$ happily
$R(n) = 40n$ , tennis-camp revenue	$\{5, 6, \ldots, 16\}$	the situation, twice — whole students, 5 to 16
$K(C) = C + 273.15$ , Celsius to Kelvin	$C \ge -273.15$	physics — nothing is colder than absolute zero

Most of the interesting restrictions in this section come from the bouncer, not the door.

The camp domain is twelve numbers, not everything between 5 and 16.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.5 — Worked example

A rectangle of fixed area

A rectangle has area $24 \text{ cm}^2$ , so its length is a function of its width: $L(w) = \dfrac{24}{w}$ . Which of $3$ , $0.5$ , $48$ , $-6$ , $0$ are possible inputs?

Candidate $w$	Possible?	Why
$3$	yes	$L(3) = 8$ . A 3 cm by 8 cm rectangle has area 24.
$0.5$	yes	$L(0.5) = 48$ . A long thin rectangle, but a real one.
$48$	yes	$L(48) = 0.5$ . The same rectangle, turned on its side.
$-6$	no	The situation alone refuses; the arithmetic would have returned $-4$ .
$0$	no	Both refuse: $\tfrac{24}{0}$ is undefined, and a rectangle with zero width is not a rectangle.

Example 1.4.8. When a value is excluded, know which of the two reasons excluded it.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.2 — What happens near the forbidden input

The gap in the graph is the gap in the domain

$x$	$1.9$	$1.999$	$2.001$	$2.1$
$f(x)$	$-60$	$-6000$	$6000$	$60$

Table 1.4.2: Outputs of $f(x) = \dfrac{6}{x-2}$ at inputs close to 2.

Creeping toward 2 from the left the outputs plunge; from the right they rocket upward. The graph splits into two pieces at $x = 2$ — the missing input, drawn.

Table 1.4.2. The excluded input is visible as a break in the curve.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.5 — Your turn

Try It Now 1.4.5

Try It Now 1.4.5

A school sells yearbooks for $25 each and printed 300 copies, so $Y(b) = 25b$ .

a) State the reasonable domain, and say which limit came from the arithmetic and which from the situation.

b) Is $260 a possible total?

Answer

a) $\{0, 1, 2, \ldots, 300\}$ . The rule $25b$ accepts every number, so the arithmetic imposes nothing; all three limits came from the situation.

b) $25b = 260$ gives $b = 10.4$ , not a whole number of yearbooks. $250 or $275, never $260.

Try It Now 1.4.5. All three limits from the situation — common for a simple multiplication model.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.6 — Domain and range from a table or a list of pairs

Here you do not reason. You read.

A coordinate pair is written $(\text{input}, \text{output})$ . So in a list of pairs the domain is every first coordinate and the range is every second coordinate — which makes this the easiest of the three representations.

List each value only once

If a function pairs $(1, 5)$ , $(2, 5)$ and $(3, 5)$ , the range is $\{5\}$ — one element. A set records which values appear, not how many times.

Order does not matter, but sorting helps

$\{4, 1, 3, 2\}$ and $\{1, 2, 3, 4\}$ are the same set. Sorting makes it far easier to read and far harder to make a mistake with.

Two conventions for writing a set down.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.6 — Worked example

Example 1.4.6: reading a domain and range off four pairs

A function is given by $(1, 3)$ , $(2, 6)$ , $(3, 12)$ , $(4, 24)$ . Write it as a table, then state the domain and the range.

Input $x$	1	2	3	4
Output $y$	3	6	12	24

Table 1.4.4: The four ordered pairs arranged as an input row and an output row.

Answer. Read the domain across the top row, $\{1, 2, 3, 4\}$ ; read the range across the bottom row, $\{3, 6, 12, 24\}$ .

Example 1.4.4. Top row is the domain, bottom row is the range.

Domain and Range · bookSHelf Integrated Math 1§1.4

Context Pause — a table is usually a sample, not the whole function

The table ran out of room; the function did not

A field trip costs $15 per student. A table lists $C(1) = 15$ , $C(2) = 30$ and $C(3) = 45$ .

The domain of that function is not $\{1, 2, 3\}$ . It is every possible number of students; the table only had room for three of them.

Read the words as well as the grid. A list of pairs is the whole function; a table of a described situation is a sample of one.

The difference between a list of pairs and a sampled table.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.6 — Your turn

Try It Now 1.4.6

Try It Now 1.4.6

a) A function is given by $(-2, 7)$ , $(0, 7)$ , $(3, 1)$ , $(5, 9)$ . State the domain and the range.

b) A table shows $P(1) = 12$ , $P(2) = 24$ , $P(3) = 36$ for pizzas at $12 each. Is the domain $\{1, 2, 3\}$ ?

Answer

a) Domain $\{-2, 0, 3, 5\}$ ; range $\{1, 7, 9\}$ — the output 7 appears twice and a set lists it once.

b) No. The words describe an ongoing situation, and nothing stops someone ordering 4 pizzas or 10. The table is a sample.

Try It Now 1.4.3. Part b is the Context Pause, tested.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.7 — Discrete and continuous

The test is one question: can the input be split?

Splits → continuous

Hours, lengths, weights and temperatures can be divided as finely as you like, so functions of those quantities are continuous and their domains are written as inequalities.

Does not split → discrete

Students, tickets, buses, pushes and barks cannot be divided, so functions of those are discrete and their domains are written as lists.

The swing gave you both at once: a height that filled a whole stretch of values, and a push count that was six separated numbers.

One question decides the type, and the type decides the notation.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.7 — Definition

Definition 1.4.4 — Discrete Function

A discrete function is a function whose domain and range consist of distinct, separate values rather than an unbroken interval of values.

The tennis-camp function is discrete. Its domain is $\{5, 6, \ldots, 16\}$ and its graph is twelve separate dots, not a line.

Definition 1.4.4. A list, and the graph is dots.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.7 — Definition

Continuous Function

Definition 1.4.5 — Continuous Function

A continuous function is a function whose graph has no breaks in it. The domain and range of a continuous function are usually described by an interval, written as an inequality.

Twelve separate dots against one unbroken segment — and the description has to match the picture.

Definition 1.4.5: twelve separate dots against one unbroken segment, and the description has to match.

Definition 1.4.5. The figure puts the camp and the garage side by side.

Domain and Range · bookSHelf Integrated Math 1§1.4

Context Pause — do not smooth over the gaps

Writing $0 \le n \le 12$ for the boxes is wrong

The Library of Congress ships books about 8 to a box, so $B(n) = 8n$ , at most 12 boxes. It claims $7.5$ and $11.9$ are inputs; there is no such thing as $7.5$ boxes. The dots have gaps, so the description must too.

	Books in boxes	Medicine in the bloodstream
Rule	$B(n) = 8n$	$M(t) = 60(0.8)^t$
Input quantity	number of boxes	hours elapsed
Can the input be split?	no	yes
Type	discrete	continuous
Domain	$\{0, 1, 2, 3, \ldots\}$	$[0, \infty)$
Range	$\{0, 8, 16, 24, \ldots\}$	$(0, 60]$

The notation is not cosmetic — it is the claim about which inputs exist. The medicine's 0 in the range gets a round bracket: the amount never actually reaches zero.

Domain and Range · bookSHelf Integrated Math 1§1.4

Insight — half a bus is not half as useful

Some quantities split. Some do not.

You can buy half a gallon of gas and get half as much driving. You cannot buy half a bus and get half as much bus.

No amount of algebra will change which side a quantity falls on. Decide it by looking at the quantity, before you write any notation down.

The discrete/continuous call is made about the world, not about the formula.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.7 — Your turn

Try It Now 1.4.7

Try It Now 1.4.7

Decide whether each function is discrete or continuous, and write its domain in the matching form.

a) A rain gauge holds up to 5 inches; $V(d)$ is the depth of water in it.

b) A theater sells tickets in packs of 4, up to 10 packs; $K(p) = 4p$ .

Answer

a) Depth is a length, so it splits: continuous, $0 \le d \le 5$ .

b) Packs do not split: discrete, $\{0, 1, 2, \ldots, 10\}$ . Writing $0 \le p \le 10$ would claim you can buy $3.5$ packs.

Try It Now 1.4.6. Same question twice, opposite answers.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.8 — Domain and range from a graph

Two motions, and they are the whole technique

Question	Direction to sweep	What you record
Domain	left to right	the smallest and largest inputs the graph reaches
Range	bottom to top	the smallest and largest outputs the graph reaches

Both rows describe the same graph, read in different directions. On a graph, domain and range become almost physical.

The two sweeps. Everything in §1.4.4 is one of these two motions.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.8 — Worked example

Example 1.4.7: sweeping a graph that doubles back

A bungee jumper's height is a function of time. The graph starts at the platform, plunges, springs back to a lower peak, and oscillates with smaller swings until the jumper is lowered. Lowest point 10 feet, highest about 75 feet, and the jump ends 35 seconds in.

Sweep left to right

$0 \le t \le 35$ The jump lasted 35 seconds.

Sweep bottom to top

$10 \le h \le 75$ Never above 75 ft, never below 10 ft.

Did they ever touch down? No — the range stops 10 feet short. The jumper passes 40 feet many times on the way down and up; the range records which heights occurred, not how often.

Example 1.4.5. The range answers a question about the jump, not about an axis.

Domain and Range · bookSHelf Integrated Math 1§1.4

Context Pause — an endpoint may just be the edge of the picture

Did the graph stop, or did the paper?

Check whether a graph really stops where it appears to, or whether it simply ran off the frame. A partial graph shows part of a function.

If a curve is still climbing when it leaves the top of the picture, you cannot read a maximum off it.

This one costs students points every year, and it costs them on the range, where a false maximum turns into a false upper bound.

A cropped graph does not report its own cropping.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.8 — Your turn

Try It Now 1.4.8

Try It Now 1.4.8

A candle burns at a steady rate. Its graph is a straight line from $(0, 20)$ down to $(8, 0)$ , and the candle is gone after that.

a) Find the domain and the range.

b) What does each one tell you about the candle?

Answer

a) $0 \le t \le 8$ hours and $0 \le h \le 20$ cm.

b) The candle burned for 8 hours before it was used up, and it was 20 cm tall when lit. Neither answer mentions an axis; both are facts about a candle.

Try It Now 1.4.4. The interpretation is the graded part.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.9 — What a graph tells you

Five features, five recipes

Feature	How to find it	Typical meaning
Vertical intercept	evaluate $f(0)$	the starting value, before anything happens
Horizontal intercept	solve $f(x) = 0$	when the quantity runs out, arrives, lands, or hits the ground
Maximum	the highest point	the largest value reached, and the input tells you when
Minimum	the lowest point	the smallest value reached, and the input tells you when
Flat stretch	a horizontal section	a period where the quantity did not change

The names matter far less than the translations in the right column.

A flat stretch means f(20) = f(22): two inputs sharing an output, which §1.3 never forbade.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.9 — Definition

Vertical Intercept

Definition 1.4.6 — Vertical Intercept

The vertical intercept is the point where the graph crosses the vertical axis. It is the output when the input is 0, so for a function $f$ it is the point $(0, f(0))$ .

It is also called the $y$ -intercept.

Definition 1.4.6: on $f(x) = 2x + 6$ each intercept carries a zero, and the zero sits in a different slot.

Definition 1.4.6. Input 0, output f(0).

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.9 — Definition

Definition 1.4.7 — Horizontal Intercept

A horizontal intercept is a point where the graph crosses the horizontal axis. It is an input whose output is 0, so it is a solution of $f(x) = 0$ . It is also called the $x$ -intercept, or a zero of the function.

A bouncing ball has many of them. Solving $h(t) = 0$ asks “at what times is the height zero?” — the first bounce, the second, and the whole stretch at the end where the ball is rolling. Each one is a moment the ball was on the ground.

Definition 1.4.7. 'A' horizontal intercept, not 'the' — there can be many.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.9 — Definition

Maximum

Definition 1.4.8 — Maximum

The maximum of a function is the greatest output value the function reaches.

A function is not required to have one. $f(x) = 2x + 6$ climbs forever to the right, so it has no greatest output at all. Dayne's maximum is 24,000 gallons, and it occurs at $x = 0$ : the maximum is the 24,000 — an output — never the 0. The input tells you when.

Definition 1.4.8: the greatest output the function reaches, and with the least output it bounds the range.

Definition 1.4.8. The kickball's 31 ft is the top of its range.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.9 — Definition

Definition 1.4.10 — Minimum

The minimum of a function is the least output value the function reaches.

Maximum and minimum are the top and bottom of the range. When you swept a graph bottom to top in §1.4.4, the two values you wrote down were the minimum and the maximum — they are not new information, they are the range’s endpoints, given names.

Definition 1.4.9. This ties §1.4.7 straight back to the sweep.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.9 — Worked example

Reading the features of $f(x) = 2x + 6$

Find the vertical intercept, the horizontal intercept, the slope, and the maximum and minimum. Every harder graph gets read the same way.

Feature	Work	Answer
Vertical intercept	$f(0) = 2(0) + 6 = 6$	$(0, 6)$
Horizontal intercept	$2x + 6 = 0 \longrightarrow x = -3$ ; check $f(-3) = 0$	$(-3, 0)$
Slope	the coefficient of $x$	2
Maximum and minimum	climbs forever right, falls forever left	neither exists

Example 1.4.10. A straight line with nonzero slope never has a maximum or a minimum.

Domain and Range · bookSHelf Integrated Math 1§1.4

Context Pause — each intercept has a zero, in a different slot

Check which coordinate the zero is sitting in

The vertical intercept above is $(0, 6)$ — input 0, output 6. The horizontal intercept is $(-3, 0)$ — input $-3$ , output 0. Swapping them turns every answer inside out. Dayne's pool, $d(x) = 24{,}000 - 1000x$ , shows what each one buys you.

Feature	Value	What it means
Vertical intercept	$(0,\ 24{,}000)$	The pool held 24,000 gallons when the pump started.
Horizontal intercept	$(24, 0)$	24 minutes in, the pool is empty — the drain is finished.
Maximum	$24{,}000$ gallons	It never held more water than it started with.
Minimum	$0$ gallons	It does get completely empty.

Every row turns a point on the graph into a fact about the pool. Whose pool was fuller to begin with? Compare vertical intercepts.

Domain and Range · bookSHelf Integrated Math 1§1.4

Insight — the maximum is the height, not the clock

A kickball’s maximum is $h(2.5) = 31$

That one statement carries two numbers with two different jobs. The output, 31, is the maximum height: about 31 feet. The input, $2.5$ , is only when it happened.

Saying the maximum was $2.5$ feet and that it happened 31 seconds in gets both numbers exactly backwards. The maximum is always an output.

The input beside a maximum tells you when, never how much.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.9 — Your turn

Try It Now 1.4.9

Try It Now 1.4.9

A candle's height in centimeters after $t$ hours is $h(t) = 20 - 2.5t$ .

a) Find the vertical intercept and say what it means.

b) Find the horizontal intercept and say what it means.

c) State the maximum and the minimum over the candle's reasonable domain.

Answer

a) $h(0) = 20 - 0 = 20$ , so the vertical intercept is $(0, 20)$ . The candle was 20 cm tall when it was lit.

b) Solve $20 - 2.5t = 0$ : $2.5t = 20$ , so $t = 8$ . The horizontal intercept is $(8, 0)$ . After 8 hours the candle has burned all the way down.

c) The reasonable domain is $[0, 8]$ — the candle does not exist before it is lit or after it is gone. Over that stretch the outputs run from 20 down to 0, so the maximum is 20 cm (at $t = 0$ ) and the minimum is 0 cm (at $t = 8$ ). The maximum is 20, not 0 — the maximum is the output, and $t = 0$ is only when it happened.

Try It Now 1.4.9. Three features, three meanings — then the maximum-is-an-output check.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.10 — Two functions on one graph

Function notation is what lets you ask about both at once

Everything so far has read one function at a time. The water park picture carries two for a reason, and §1.3’s notation is what lets you ask about both at once.

Question about the two pools	What notation asks	Direction
How much is in Aly's pool at 5 minutes?	find $a(5)$	input given, output wanted
When is Dayne's pool down to 2000 gallons?	solve $d(x) = 2000$	output given, input wanted — the backwards question
When do the pools hold the same amount?	solve $a(x) = d(x)$	where the graphs cross
When does Aly's pool hold more?	solve $a(x) > d(x)$	where one segment is above the other

The crossing is what the picture advertised: at some point both pools hold the same amount.

Four questions, two directions. The input at which two graphs cross is the solution of the equation that sets them equal.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.10 — Worked example

Example 1.4.9: four questions about two pools

Using $a(x) = 28{,}000 - 1400x$ and $d(x) = 24{,}000 - 1000x$ , answer four questions.

Question	Work	Answer
Find $a(5)$	$a(5) = 28{,}000 - 1400(5) = 28{,}000 - 7000 = 21{,}000$	21,000 gal — time in, amount out.
Solve $d(x) = 2000$	$24{,}000 - 1000x = 2000 \Rightarrow x = 22$	$x = 22$ minutes
Solve $a(x) = d(x)$	$28{,}000 - 1400x = 24{,}000 - 1000x \Rightarrow 4000 = 400x$	$x = 10$ ; check $d(10) = a(10) = 14{,}000$
Solve $a(x) > d(x)$	Above until the crossing	$0 \le x < 10$

Answer. A question about two functions answers with an interval of inputs, not a number.

Example 1.4.9. Reading the crossing off the picture and solving the equation on paper are the same question asked two ways.

Domain and Range · bookSHelf Integrated Math 1§1.4

§1.4.10 — Your turn

Try It Now 1.4.10

Try It Now 1.4.10

Two candles are lit at the same time. One is $h(t) = 20 - 2.5t$ and the other is $g(t) = 30 - 5t$ , both in centimeters after $t$ hours.

a) Which is taller at the moment they are lit?

b) When are they the same height?

c) Which burns out first?

Answer

a) Evaluate both at $t = 0$ : $h(0) = 20$ and $g(0) = 30$ . The second candle starts taller.

b) Set them equal: $20 - 2.5t = 30 - 5t$ , so $2.5t = 10$ and $t = 4$ . Check: $h(4) = 10$ and $g(4) = 10$ . Both are 10 cm tall after 4 hours.

c) $h(t) = 0$ gives $t = 8$ ; $g(t) = 0$ gives $t = 6$ . The second candle burns out first, at 6 hours, even though it started 10 cm taller — it burns twice as fast.

Try It Now 1.4.10. The water park again with different numbers: the one that starts higher and falls faster is overtaken partway through and finishes first.

Domain and Range · bookSHelf Integrated Math 1§1.4

Key Terminology — the nine words this section defines

Key terms

domain — the set of all possible input values of a function.

range — the set of all possible output values of a function.

reasonable domain — the set of inputs that make sense in the situation, usually smaller than the set the rule alone would accept.

discrete function — a function whose domain and range are distinct, separate values rather than an unbroken interval.

continuous function — a function whose graph has no breaks, so its domain and range are described by intervals.

vertical intercept — where a graph crosses the vertical axis, at $(0, f(0))$ ; also called the $y$ -intercept.

horizontal intercept — where a graph crosses the horizontal axis, a solution of $f(x) = 0$ ; also called the $x$ -intercept.

maximum — the greatest output value a function reaches, which is the top of its range.

minimum — the least output value a function reaches, which is the bottom of its range.

Second column reveals on click.

Domain and Range · bookSHelf Integrated Math 1§1.4

The headline result

The situation’s domain, not the rule’s, is the one the model is about

The rule $d(x) = 24{,}000 - 1000x$ accepts every real number there is. Dayne's pool accepts the ones from 0 through 24. Nothing in the arithmetic knows the difference — $d(30) = -6000$ claims a pool containing negative six thousand gallons, and there is no such thing. The rule is not wrong; it is answering a question nobody should have asked it. The whole gap between a formula and a model lives in that comparison.

\text{rule: all real } x \qquad \longrightarrow \qquad \text{situation: } [0,\, 24]

† A model is a rule plus the set of inputs on which the rule means something, and the second half is supplied by the situation.

One headline: two answers, and only one of them is about the world.

Domain and Range · bookSHelf Integrated Math 1§1.4

1.4

§1.4 — Conclusions

What to carry forward

The one idea

Ask two questions of every model, not one: what would the rule accept, and what does the situation permit? On a graph the answers are two sweeps — left to right for the domain, bottom to top for the range — and the endpoints you write down are the minimum and the maximum, given names.

Where it goes wrong

Writing a discrete domain as an interval; reading a maximum off a graph that only ran out of paper; treating a sampled table as the whole function; and quoting a maximum’s input as if it were the output.

Next: §1.5 changes subject to exponents. Keep the domain question in your pocket — a negative exponent puts a quantity underneath a fraction bar, which reopens exactly the question you just learned to ask. Back to start.

The failure list is the four defects this section is written to prevent.

Domain and Range

Objectives

Two pumps, two pools, one pair of axes

Dayne's pump: building a rule from a rate

Aly's pump: building a rule from two points on a graph

Try It Now 1.4.1

Not “all numbers” — all the possible ones

Domain

Try It Now 1.4.1

The range follows from the domain

Range

Two range questions, one method

Ranges that surprise you

Try It Now 1.4.2

A shorthand for “every time from here to there”

Interval Notation

Try It Now 1.4.4

Two domains compete for the name

Same thirty seconds, four different ranges

Reasonable Domain

Three functions, three different reasons

A rectangle of fixed area

The gap in the graph is the gap in the domain

Try It Now 1.4.5

Here you do not reason. You read.

List each value only once

Order does not matter, but sorting helps

Example 1.4.6: reading a domain and range off four pairs

The table ran out of room; the function did not

Try It Now 1.4.6

The test is one question: can the input be split?

Splits → continuous

Does not split → discrete

Continuous Function

Writing 0≤n≤120 \le n \le 120≤n≤12 for the boxes is wrong

Some quantities split. Some do not.

Try It Now 1.4.7

Two motions, and they are the whole technique

Example 1.4.7: sweeping a graph that doubles back

Sweep left to right

Sweep bottom to top

Did the graph stop, or did the paper?

Try It Now 1.4.8

Five features, five recipes

Vertical Intercept

Maximum

Reading the features of f(x)=2x+6f(x) = 2x + 6f(x)=2x+6

Check which coordinate the zero is sitting in

A kickball’s maximum is h(2.5)=31h(2.5) = 31h(2.5)=31

Try It Now 1.4.9

Function notation is what lets you ask about both at once

Example 1.4.9: four questions about two pools

Try It Now 1.4.10

Key terms

What to carry forward

The one idea

Where it goes wrong

Writing $0 \le n \le 12$ for the boxes is wrong

Reading the features of $f(x) = 2x + 6$

A kickball’s maximum is $h(2.5) = 31$