Domain and Range · bookSHelf Integrated Math 1§1.4

Do Now

Recall from §1.3: a function pairs each input with exactly one output.

A sunflower is measured every few days after it sprouts. Below, the number of days since it sprouted is paired with its height in centimetres on that day.

days since it sproutedheight (cm)
02
49
7.516

Is this a table of a function? How do you know?

Two arrows into one output is a function; two arrows out of one input is not Two mapping diagrams sit side by side under the line every input has exactly one output. In the left diagram an input oval holds 1, 2, 3 and 4 and an output oval holds 7, 9 and 12. Arrows run from 1 to 7, from 2 to 9, from 3 to 7, and from 4 to 12. The two arrows arriving at 7, one from 1 and one from 3, are picked out in blue and the output 7 is ringed, and the panel is labelled a function, two different inputs sharing an output, fine. In the right diagram an input oval holds 1, 2 and 4 and an output oval holds 5, 7, 9 and 12. Arrows run from 1 to 7, from 2 to 9, from 2 to 5, and from 4 to 12. The two arrows leaving the input 2, one to 9 and one to 5, are picked out in rust and the input 2 is ringed, and the panel is labelled not a function, one input with two different outputs, fatal. A closing line reads: the picture is worth carrying, many arrows into one output is fine; two arrows out of one input is not. every input has exactly one output inputs outputs 1 2 3 4 7 9 12 a function Two different inputs sharing an output: fine. inputs outputs 1 2 4 5 7 9 12 not a function One input with two different outputs: fatal. The picture is worth carrying: many arrows into one output is fine; two arrows out of one input is not.

Is this a table of a function? How do you know?

Bridge line — say this once they confirm it

You just confirmed these are functions. Today: which inputs are even allowed?

bookSHelf · Integrated Math 11 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

The situation

Aly and Dayne work at a water park. At the end of each month they have to drain the small pool at the bottom of the ride they supervise, each with a pump. One graph shows both drains at once: Aly's pool follows a(x)=28,000−1400xa(x) = 28{,}000 - 1400x, and Dayne's follows d(x)=24,000−1000xd(x) = 24{,}000 - 1000x, both in gallons against the minutes xx since the pumps switched on. Before any arithmetic, look at the picture and say what you see.
Two pools draining on one pair of axes, crossing once A graph of water remaining against time for two pools at a water park. The horizontal axis is time in minutes, the vertical axis is water in gallons. Aly's pool starts at 28,000 gallons and falls in a straight line to zero at 20 minutes. Dayne's pool starts lower, at 24,000 gallons, and falls less steeply in a straight line to zero at 24 minutes. Because Aly's line is steeper it overtakes Dayne's: the two segments cross exactly once, at 10 minutes, where both pools hold 14,000 gallons, and after that Aly's line runs below Dayne's and reaches the axis first. A dot marks the crossing, with dashed guides down to 10 on the time axis and across to 14,000 on the gallons axis. A closing line reads: at some point in time, both pools will have the same amount of water. Both pools are draining. Watch which one starts higher, which falls faster, and where they meet. water (gallons) time (minutes) 0 Aly a(x) = 28,000 − 1400x Dayne d(x) = 24,000 − 1000x 28,000 24,000 10 10 minutes in, both hold 14,000 gallons 20 24 At some point in time, both pools will have the same amount of water.
bookSHelf · Integrated Math 12 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Predict — then read it off

1. Read how many gallons are left in Aly’s pool at x = 5 minutes. Write it down.

2. Read Aly’s gallons at x = 15 minutes. Write it down.

3. Read Aly’s gallons at x = 22 minutes. Write it down.

What went wrong on the third one?

Two pools draining on one pair of axes, crossing once A graph of water remaining against time for two pools at a water park. The horizontal axis is time in minutes, the vertical axis is water in gallons. Aly's pool starts at 28,000 gallons and falls in a straight line to zero at 20 minutes. Dayne's pool starts lower, at 24,000 gallons, and falls less steeply in a straight line to zero at 24 minutes. Because Aly's line is steeper it overtakes Dayne's: the two segments cross exactly once, at 10 minutes, where both pools hold 14,000 gallons, and after that Aly's line runs below Dayne's and reaches the axis first. A dot marks the crossing, with dashed guides down to 10 on the time axis and across to 14,000 on the gallons axis. A closing line reads: at some point in time, both pools will have the same amount of water. Both pools are draining. Watch which one starts higher, which falls faster, and where they meet. water (gallons) time (minutes) 0 Aly a(x) = 28,000 − 1400x Dayne d(x) = 24,000 − 1000x 28,000 24,000 10 10 minutes in, both hold 14,000 gallons 20 24 At some point in time, both pools will have the same amount of water.

Predict → Observe → Explain. The hold is the mechanism.

Let all three be attempted before anyone speaks. The first two succeed, which is what makes the third one land — two confident readings, then nothing to read. Do not rescue it. Ask what they wrote, not what the answer is. Someone will say “it isn’t there” or “it stops”; that sentence is the whole lesson and it belongs to them.

Do not say the word “domain” here, and do not accept it if a student who has seen it before offers it. The term arrives after they have coined their own.

bookSHelf · Integrated Math 13 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Investigate — Contrast

Aly's line stops at x = 20. Dayne's line, on the very same pair of axes, keeps going to x = 24. Same starting instant, same picture, two different stopping points. Why don't the two functions have to stop at the same time?

Steer toward: each pool's own pump decides where its line stops — Aly's finishes at 20 minutes, Dayne's at 24 — and sharing one picture does not force the two functions to share a stopping point.

bookSHelf · Integrated Math 14 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Investigate — Separation

Hold Dayne's rate and starting amount fixed — 24,000 gallons, 1000 gallons a minute. Only change how long the park lets the ride run before shutting the drain off early: first the usual 24 minutes, then just 15 minutes, well before the pool is even empty. What changes about the graph? What stays exactly the same?

Steer toward: only where the line stops — the width of the band along the bottom — moves. The straight-line shape and the starting amount, 24,000 gallons, don't change at all.

bookSHelf · Integrated Math 15 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Investigate — Generalisation

New situation, same question: a table at a fundraiser sells raffle tickets from the moment it opens until it closes for the day. Which input values — number of tickets sold so far — are allowed here?

timetickets sold so far
9:00 am (table opens)0
1:00 pm62
6:00 pm (table closes)210

Steer toward: whole numbers, starting at 0, up to whatever the largest count was when the table closed — the same left-to-right sweep idea, on a completely different picture.

bookSHelf · Integrated Math 16 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Investigate — Fusion

Now restrict both what's allowed to go IN and what can come OUT, at the same time, for Aly's pool alone: only count the minutes before her pool drops below 10,000 gallons, and before 15 minutes have passed. What changes along the bottom of the graph? What changes along the side?

Steer toward: two separate bands, read independently — one along the time axis, one along the gallon axis. Restricting the gallons does not automatically restrict the time, or the other way around.

bookSHelf · Integrated Math 17 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Hinge check

The water park graph runs Aly's line from x = 0 to x = 20 minutes along the bottom. Which values could the TIME actually take?

  1. A. Only x = 0 and x = 20
  2. B. Every value from 0 to 20
  3. C. Only whole-number minutes from 0 to 20
  4. D. Every value from 0 to 28,000

Hinge check — answer key

A. Only x = 0 and x = 20 — misconception: endpoints-for-interval — reads a continuous span as just its two endpoint values

B. Every value from 0 to 20 — CORRECT

C. Only whole-number minutes from 0 to 20 — misconception: unwarranted discreteness — restricts a continuous input to integers with no reason from the situation

D. Every value from 0 to 28,000 — misconception: axis-swap — answers with the graph's gallon values instead of its time values

Decision: ≥80% B → proceed to reveal. 50–80% → re-teach endpoints-for-interval (A) directly against the graph's solid band. <50% → re-run Investigate from Contrast.

bookSHelf · Integrated Math 18 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Name It

In the Investigate run you decided which times the water park graph actually reaches, and which gallon amounts it actually reaches. The point of math is to do that again for any graph. Write down, in your own words, exactly what your group did — then name and define BOTH methods.

  • Which direction did you sweep first — left to right, or bottom to top?
  • What were you looking for as you swept?
  • How did you decide where each sweep started and stopped?
  • Did the two sweeps give you the same KIND of answer, or two different kinds?

Now name what you found, and define it in one sentence — as if it were going in a textbook.

A student might land on something like — The Sideways Set, every input value the graph actually reaches, found by sweeping left to right; and The Up-and-Down Set, every output value the graph actually reaches, found by sweeping bottom to top.

This IS the naming act — the textbook’s word comes next, not now

Silent individual writing first, then pairs. Do not narrate the sweeping procedure for them — producing it is the task. Write two or three coined names on the board verbatim, with the students’ names attached. Reveal the sample coinages only if the room is genuinely stuck, and reveal them as one group’s answer rather than as the right answer.

If a team writes a definition that is wrong but self-consistent, leave it standing. The formal reveal on the next slide is what corrects it, and the correction lands harder against something they committed to.

bookSHelf · Integrated Math 19 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

The Question Driving §1.4

Which inputs is a function actually allowed to take, and which outputs does it actually reach?

You just pinned down where ONE graph's two sweeps started and stopped. Here's the question that drives the rest of §1.4 — and by the end you'll have a precise name for every way that answer can be shaped, restricted, or run out.

bookSHelf · Integrated Math 110 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

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 1.4.2 — 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.

Consolidation — reference what the class actually tried

Some of you swept left to right along the bottom and read where Aly's pool started and stopped; others swept bottom to top and read the lowest and highest gallon amount the pool held. That split is exactly the difference between the inputs the graph accepts and the outputs it produces.

bookSHelf · Integrated Math 111 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

The picture

Sweeping left to right lays down the band of times the graph reaches, and that band is the domain A graph of a bungee jumper's height in feet against time in seconds. The curve starts at 75 feet at time 0, plunges to 10 feet, springs back to a lower peak, and keeps oscillating with smaller and smaller swings before levelling off near thirty-two feet, where the stretched cord holds the jumper until the ride ends at 35 seconds. A vertical line then sweeps from the left edge of the graph to the right edge, and as it travels it lays down a solid band along the horizontal axis running from 0 to 35. The band is labelled: the domain, every input the graph reaches. Below the horizontal axis the inequality 0 is less than or equal to t is less than or equal to 35 appears, and a closing line reads: the domain says the jump lasted 35 seconds. In a bungee jump, the height of the jumper is a function of the time since the jump began. height (feet) 75 10 0 35 time (seconds) sweep left to right the domain — every input the graph reaches 0 ≤ t ≤ 35 The domain says the jump lasted 35 seconds.
bookSHelf · Integrated Math 112 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

The picture

Sweeping bottom to top lays down the band of heights the graph reaches, and that band is the range The same graph of a bungee jumper's height in feet against time in seconds used for the domain. A horizontal line sweeps upward from the bottom of the graph. Nothing is collected until it reaches 10 feet; from there a solid band grows upward along the vertical axis and stops at 75 feet, because the graph never goes higher. The band is labelled with the inequality 10 is less than or equal to h is less than or equal to 75. A dashed line is then drawn across the graph at 40 feet and a dot is placed at each of the seven places the curve crosses it, above the lines: the jumper passes through a height of 40 feet many times, and it records which heights occurred, not how often. A closing line reads: the range stops 10 feet short. Domain is about how wide the graph is. Range is about how tall it is. height (feet) 75 10 0 35 time (seconds) sweep bottom to top 10 ≤ h ≤ 75 40 The jumper passes through a height of 40 feet many times. It records which heights occurred, not how often. The range stops 10 feet short.
bookSHelf · Integrated Math 113 / 30

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.

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

bookSHelf · Integrated Math 114 / 30
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

Critique this claim

Jordan thinks Aly's domain must be the same as Dayne's, since both pools are drawn on the very same pair of axes. Is Jordan correct? Why or why not?

Two pools draining on one pair of axes, crossing once A graph of water remaining against time for two pools at a water park. The horizontal axis is time in minutes, the vertical axis is water in gallons. Aly's pool starts at 28,000 gallons and falls in a straight line to zero at 20 minutes. Dayne's pool starts lower, at 24,000 gallons, and falls less steeply in a straight line to zero at 24 minutes. Because Aly's line is steeper it overtakes Dayne's: the two segments cross exactly once, at 10 minutes, where both pools hold 14,000 gallons, and after that Aly's line runs below Dayne's and reaches the axis first. A dot marks the crossing, with dashed guides down to 10 on the time axis and across to 14,000 on the gallons axis. A closing line reads: at some point in time, both pools will have the same amount of water. Both pools are draining. Watch which one starts higher, which falls faster, and where they meet. water (gallons) time (minutes) 0 Aly a(x) = 28,000 − 1400x Dayne d(x) = 24,000 − 1000x 28,000 24,000 10 10 minutes in, both hold 14,000 gallons 20 24 At some point in time, both pools will have the same amount of water.

No — Aly's line stops at x=20 while Dayne's keeps going to x=24; sharing one picture does not force two functions to share a domain. Read each curve's own extent separately.

# misconception: shared-axes-implies-shared-domain — treats two functions drawn on one picture as if they must share the same allowed inputs, rather than reading each curve's own extent.

bookSHelf · Integrated Math 115 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Application — 1 of 2

Dayne's pool follows d(x)=24,000−1000xd(x) = 24{,}000 - 1000x. Find the domain and the range.

Domain 0 ≤ x ≤ 24 minutes (starts at x=0, d(24)=0). Range 0 ≤ d ≤ 24,000 gallons.

bookSHelf · Integrated Math 116 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Application — 2 of 2

A candle burns at a steady rate. Its height in centimeters is a function of the time in hours since it was lit. The graph is a straight line from (0, 20) down to (8, 0), and the candle is gone after that: h(t)=20−2.5th(t) = 20 - 2.5t. Find the domain and the range.

Domain 0 ≤ t ≤ 8 hours. Range 0 ≤ h ≤ 20 cm.

bookSHelf · Integrated Math 117 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

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=0x = 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=24x = 24 the pool holds 0 gallons, which is a real reading too.

Consolidation — reference what the class actually tried

Some of you wrote a ‘≤’ on both sides out of habit; others reached for a bracket without being sure which shape belongs at which end. That split is exactly why the bracket needs a precise rule, checked endpoint by endpoint against the situation, rather than picked by feel.

Two minutes. This one is told, not discovered. The motivator slide that used to sit ahead of this one is now a spoken sentence: name the scenario the class already solved, say what the previous definition could not describe about it, then put this definition up. Ten definitions cannot all be discovered in one period — the anchor got the full cycle, these get named and practised.

bookSHelf · Integrated Math 118 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Application

Write Aly's domain and range in interval notation.

Domain [0,20][0, 20]. Range [0, 28,000][0,\ 28{,}000].

bookSHelf · Integrated Math 119 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

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 2.52.5 buses. A rocket’s height stops meaning anything once it lands.

The rule accepts an unbroken number line; the situation clips it three times, down to the whole numbers from 0 through 100 The rule T of n equals 7 n is stated above a note that tickets are 7 dollars each and the theater has 100 seats. Under the heading what would the rule accept, an unbroken number line runs off the page in both directions, noted as any number at all, since 7 times negative 3 is negative 21 and 7 times 2.5 is 17.5, and the arithmetic never complains. Under the heading what does the situation permit, the same unbroken line appears and is then clipped three times. First the unbroken line is replaced by separated dots, one at each whole number, because whole numbers only, you cannot sell 2.5 tickets. Then the dots to the left of 0 and the arrow beyond them disappear, at least 0, since you cannot sell a negative number of tickets. Then the dots to the right of 100 and the arrow beyond them disappear, at most 100, since the theater has 100 seats. What survives is a bracketed run of dots from 0 to 100, written as the set 0, 1, 2, and so on to 100, with a closing line: it is the whole numbers from 0 through 100. T(n) = 7n Tickets are $7 each, and the theater has 100 seats. What would the rule accept? Any number at all. 7 · (−3) = −21; 7 · 2.5 = 17.5. The arithmetic never complains. What does the situation permit? 0 25 50 75 100 Whole numbers only — you cannot sell 2.5 tickets. At least 0, since you cannot sell a negative number of tickets. At most 100, since the theater has 100 seats. {0, 1, 2, …, 100} It is the whole numbers from 0 through 100.

Consolidation — reference what the class actually tried

Some of you kept every number the rule would compute; others started crossing values off because of what the water park situation actually allows. That split is exactly the difference between what the arithmetic accepts and what the situation permits — and the second one is the reasonable domain.

Two minutes. This one is told, not discovered. The motivator slide that used to sit ahead of this one is now a spoken sentence: name the scenario the class already solved, say what the previous definition could not describe about it, then put this definition up. Ten definitions cannot all be discovered in one period — the anchor got the full cycle, these get named and practised.

bookSHelf · Integrated Math 120 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Application

For d(x)=24,000−1000xd(x) = 24{,}000 - 1000x, state what the rule alone would accept, then find the reasonable domain. Is x = -5 reasonable? Is x = 18?

Rule alone: every real number. Reasonable domain 0≤x≤240 \le x \le 24. x = -5 is not reasonable (before the pump started). x = 18 is reasonable — d(18) = 6000 gallons, still well within the park's open hours.

bookSHelf · Integrated Math 121 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

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.

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.

Twelve separate dots against one unbroken segment: the difference between a discrete and a continuous function Two graphs side by side, separated by a vertical rule. On the left is the tennis camp, R of n equals 40 n, whose input is a number of students and cannot be split; twelve separate dots appear one at a time, from 5 students at 200 dollars up to 16 students at 640 dollars, and the gaps between them stay empty. On the right is the parking garage, P of h equals 5 h plus 10, whose input is hours parked and can be split as finely as you like; a single straight segment is drawn in one unbroken stroke from 0 hours at 10 dollars to 6 hours at 40 dollars. Beneath the left graph: twelve separate points, the set 5, 6, 7 and so on to 16, discrete. Beneath the right graph: one unbroken segment, 0 is less than or equal to h is less than or equal to 6, continuous. A closing line reads: the dots have gaps between them, so the description has to have gaps too. The test is nearly always the same one question: can the input quantity be split? Tennis camp R(n) = 40n number of students — cannot be split $200 $640 5 16 Parking garage P(h) = 5h + 10 hours parked — can be split as finely as you like $10 $40 0 6 12 separate points one unbroken segment {5, 6, 7, …, 16} 0 ≤ h ≤ 6 discrete continuous The dots have gaps between them, so the description has to have gaps too.

Consolidation — reference what the class actually tried

Some of you focused on how many boxes a shipment could hold, a fixed count; others focused on whether you could land in between two of them. That split is exactly the difference between a function that's discrete, because its inputs are separate, countable steps, and one that's continuous, because its inputs fill in every gap between them.

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

Two minutes. This one is told, not discovered. The motivator slide that used to sit ahead of this one is now a spoken sentence: name the scenario the class already solved, say what the previous definition could not describe about it, then put this definition up. Ten definitions cannot all be discovered in one period — the anchor got the full cycle, these get named and practised.

bookSHelf · Integrated Math 122 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Application

Classify each and write its domain in the matching form: (a) books arriving in boxes, B(n)=8nB(n) = 8n, for a shipment of at most 12 boxes. (b) medicine leaving a bloodstream, M(t)=60(0.8)tM(t) = 60(0.8)^t, observed for the first 5 hours.

(a) Discrete, {0,1,…,12}\{0, 1, \ldots, 12\} — writing 0≤n≤120 \le n \le 12 would be wrong, since it claims 7.5 boxes is allowed. (b) Continuous, 0≤t≤50 \le t \le 5.

bookSHelf · Integrated Math 123 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

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 ff it is the point (0,f(0))(0, f(0)).

It is also called the yy-intercept.

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)=0f(x) = 0. It is also called the xx-intercept, or a zero of the function.

On f(x) = 2x + 6 the zero sits in a different slot in each intercept A pair of axes with the straight line f of x equals 2 x plus 6 drawn across them. Where the line crosses the vertical axis a dot is placed and labelled the point (0, 6), worked out as f of 0 equals 2 times 0 plus 6 equals 6. Where the line crosses the horizontal axis a second dot is placed and labelled the point (negative 3, 0), worked out by solving 2 x plus 6 equals 0 to get x equals negative 3. Below the graph the two points are written out side by side with the zero boxed in each: under vertical intercept, the pair (0, 6) with the first slot boxed, input 0, output 6; under horizontal intercept, the pair (negative 3, 0) with the second slot boxed, input negative 3, output 0. A closing line reads: swapping them turns every answer inside out. f(x) = 2x + 6 x y f(0) = 2(0) + 6 = 6 (0, 6) (−3, 0) 2x + 6 = 0 → x = −3 vertical intercept (0, 6) input 0, output 6 horizontal intercept (−3, 0) input −3, output 0 Swapping them turns every answer inside out.

Consolidation — reference what the class actually tried

Some of you found the point where the INPUT was 0; others found the point where the OUTPUT was 0. That split is exactly the difference between a vertical intercept and a horizontal intercept — the zero just sits in a different slot.

A bouncing ball has many of them. Solving h(t)=0h(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.

Two minutes. This one is told, not discovered. The motivator slide that used to sit ahead of this one is now a spoken sentence: name the scenario the class already solved, say what the previous definition could not describe about it, then put this definition up. Ten definitions cannot all be discovered in one period — the anchor got the full cycle, these get named and practised.

bookSHelf · Integrated Math 124 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Application

Dayne's pool has vertical intercept (0, 24,000) and horizontal intercept (24, 0). Aly's has vertical intercept (0, 28,000) and horizontal intercept (20, 0). Interpret each of Dayne's two facts in words, then say which pool started fuller and which finished first.

(0, 24,000): the pool held 24,000 gallons when the pump started. (24, 0): the pool is completely empty at 24 minutes. Comparing vertical intercepts, Aly's pool started fuller (28,000 > 24,000). Comparing horizontal intercepts, Aly's pool finished first — her intercept, 20, is smaller than Dayne's, 24.

bookSHelf · Integrated Math 125 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

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+6f(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=0x = 0: the maximum is the 24,000 — an output — never the 0. The input tells you when.

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.

The maximum and the minimum are the two ends of the range, given names A graph of a kickball's height in feet against time in seconds, kicked from ground level. The curve starts at the ground, climbs to a peak, and comes back down to the ground. A dot marks the peak, with dashed guides dropped from it to both axes: across to 31 on the height axis and down to 2.5 on the time axis, labelled the maximum is 31 ft, the output, and reached at 2.5 s, the input. A second dot marks the landing point, labelled the lowest output is 0, at landing. Marks on the height axis at 0 and at 31 are then read together as the range, 0 is less than or equal to h is less than or equal to 31. A closing line reads: they are not new information, they are the range's endpoints, given names. A kickball is kicked from ground level; its height is a function of the time since the kick. height (feet) 0 5 time (seconds) 31 2.5 the maximum is 31 ft — the output reached at 2.5 s — the input the lowest output is 0, at landing range 0 ≤ h ≤ 31 They are not new information — they are the range's endpoints, given names.

Consolidation — reference what the class actually tried

Some of you looked for the highest POINT on the kickball's flight and found one; others tried the same search on the line and couldn't, no matter how far right you traced it. That split is exactly the difference between a function that has a maximum and one that doesn't — because the line never stops climbing.

Two minutes. This one is told, not discovered. The motivator slide that used to sit ahead of this one is now a spoken sentence: name the scenario the class already solved, say what the previous definition could not describe about it, then put this definition up. Ten definitions cannot all be discovered in one period — the anchor got the full cycle, these get named and practised.

bookSHelf · Integrated Math 126 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Application

A kickball is kicked from ground level. Its maximum is h(2.5) = 31 (about 31 feet, 2.5 seconds after the kick). Interpret this fact in words, then state what you can say about the flight's minimum.

The ball reaches its greatest height, about 31 feet, 2.5 seconds after the kick. Its minimum is 0 feet, ground level — reached both at the instant it's kicked (t=0) and again when it lands.

bookSHelf · Integrated Math 127 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Checkpoint

Sort each item into its bucket. Nothing to compute — just classify.

  1. A dog’s medicine level can settle at any reading as the hours pass. — discrete or continuous?
  2. A shipment log can only ever hold a whole number of boxes. — discrete or continuous?
  3. “The pump only runs for at most 24 minutes.” — a fact about the domain, or the range?
  4. “The pool never holds more than 24,000 gallons.” — a fact about the domain, or the range?
  5. “x cannot equal 2, because the denominator would be 0.” — caused by the rule’s arithmetic, or by the situation?
  6. “n cannot be negative, because you can’t sell a negative number of tickets.” — arithmetic, or situation?
  7. The point (0, 40) on a graph. — vertical intercept, or horizontal intercept?
  8. The point (8, 0) on a graph. — vertical intercept, or horizontal intercept?

Answer key

1 continuous. 2 discrete. 3 domain. 4 range. 5 arithmetic. 6 situation. 7 vertical. 8 horizontal.

Items 7 and 8 are click-to-reveal and test intercepts — run them only if you taught the 1.4.7/1.4.8 extension. Items 1–6 cover the core path on their own.

Whole-class, hands or mini-whiteboards, no writing time. If this runs past three minutes the five refinements did not land, and the fix is next lesson, not more minutes here.

bookSHelf · Integrated Math 128 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

The Question, Answered

Which inputs is a function actually allowed to take, and which outputs does it actually reach?

The domain is the first: every input the function actually accepts. The range is the second: every output it actually reaches. Everything else in this section was about describing the exact shape of those two sets, one precise name at a time:

question about the domain or rangeanswered by
What does it actually reach?Domain, Range
Separate steps, or an unbroken stretch?Discrete, Continuous
What does the situation rule out?Reasonable Domain
Where does it cross each axis?Vertical, Horizontal Intercept
How high or low does it actually go?Maximum, Minimum
bookSHelf · Integrated Math 129 / 30
Domain and Range · bookSHelf Integrated Math 1§1.4

Exit Ticket

Silently, from memory, no notes: write your own definition of each.

  1. Domain
  2. Discrete function
  3. Maximum
bookSHelf · Integrated Math 130 / 30