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

bookSHelf · Integrated Math 11 / 25
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

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.

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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=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.

Square bracket if the endpoint is in the set, round if it is not Four number lines, one above another, each mapped to its interval notation. The first runs from a filled endpoint at 0 to a filled endpoint at 24, squared on both sides and labelled open bracket 0 comma 24 close bracket, both endpoints included. The second runs from an open endpoint at 0 to an open endpoint at 24, rounded on both sides and labelled with round brackets, neither endpoint included. The third runs from a filled endpoint at 0 to an open endpoint at 24, one of each, 0 is included, 24 is not. The fourth starts filled at 0 and runs off the page in an arrow, labelled open bracket 0 comma infinity close bracket, with a closing note: infinity is a direction, not a number, so it always takes a round bracket. Square bracket where the endpoint is included; round where it is not. 0 24 [0, 24] both endpoints included 0 24 (0, 24) neither endpoint included 0 24 [0, 24) 0 is included, 24 is not 0 ∞ [0, ∞) 0 and everything above it, forever Infinity is a direction, not a number, so it always takes a round bracket.

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

bookSHelf · Integrated Math 13 / 25
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

§1.4.5 — 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 2.52.5 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.

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

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Three clips: whole units, a floor, and a ceiling.
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,…,16}\{5, 6, \ldots, 16\} and its graph is twelve separate dots, not a line.

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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.

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.

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

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

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.

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.
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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+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.

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.

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

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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.

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Definition 1.4.9. This ties §1.4.7 straight back to the sweep.
Domain and Range · bookSHelf Integrated Math 1§1.4

The situation

Build a table for f(x)=6x−2f(x) = \dfrac{6}{x - 2} at x=−10,0,1,2,8x = -10, 0, 1, 2, 8, and describe what the graph does near the excluded value.

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Domain and Range · bookSHelf Integrated Math 1§1.4

Worked example — Step

Step 1 — the table.

xx −10-10 00 11 22 88
f(x)f(x) −0.5-0.5 −3-3 −6-6 ? 11

Check two of them: f(−10)=6−10−2=6−12=−0.5f(-10) = \dfrac{6}{-10 - 2} = \dfrac{6}{-12} = -0.5, and f(8)=68−2=66=1f(8) = \dfrac{6}{8 - 2} = \dfrac{6}{6} = 1.

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Domain and Range · bookSHelf Integrated Math 1§1.4

Worked example — Step

Step 2 — the fifth entry. When x=2x = 2 the denominator becomes 2−2=02 - 2 = 0, and the rule asks you to compute 60\dfrac{6}{0}. There is no such number, and a calculator will not produce one either — it returns an error, because there is nothing to return. So the domain is all real numbers except 2, written x≠2x \neq 2.

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Domain and Range · bookSHelf Integrated Math 1§1.4

Worked example — Step

Step 3 — creep up on the forbidden input from both sides.

xx 1.81.8 1.91.9 1.951.95 1.9991.999 2.0012.001 2.052.05 2.12.1 2.22.2
f(x)f(x) −30-30 −60-60 −120-120 −6000-6000 60006000 120120 6060 3030

As xx creeps toward 2 from the left the outputs plunge; as it creeps toward 2 from the right they rocket upward instead.

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Domain and Range · bookSHelf Integrated Math 1§1.4

Worked example — Answer

Answer: The domain is x≠2x \neq 2, and the graph is not one connected curve — it splits into two pieces at x=2x = 2. The gap in the graph is the gap in the domain, drawn.

This is also the first domain in the section that no interval can express. "All real numbers except 2" is a stretch of the number line with a hole punched in it, and an interval has no way to punch a hole.

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Domain and Range · bookSHelf Integrated Math 1§1.4

The situation

Read the domain and range of the bungee jump from its graph, and say what each one means about the jump.

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Domain and Range · bookSHelf Integrated Math 1§1.4

Worked example — Step

Step 1 — domain, sweeping left to right. The leftmost time on the graph is t=0t = 0, the instant of the jump; the rightmost is t=35t = 35, when it ends. So the domain is [0,35][0, 35], or 0≤t≤350 \le t \le 35.

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Domain and Range · bookSHelf Integrated Math 1§1.4

Worked example — Step

Step 2 — range, sweeping bottom to top. The lowest height the graph ever reaches is 10 feet — the bottom of the first plunge, the closest the jumper comes to the river. The highest is about 75 feet, at the platform. So the range is [10,75][10, 75].

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Domain and Range · bookSHelf Integrated Math 1§1.4

Worked example — Step

Step 3 — read them back into the situation. The domain says the jump lasted 35 seconds. The range says the jumper was never higher than 75 feet and never lower than 10 feet above the water — which is also the answer to "did they touch the river?" No; the range stops 10 feet short.

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Domain and Range · bookSHelf Integrated Math 1§1.4

Worked example — Answer

Answer: domain [0,35][0, 35], range [10,75][10, 75].

Now notice what the sweep ignores. The jumper passes through a height of 40 feet many times — on the way down, on the way up, on the way down again. The range does not care. It records which heights occurred, not how often. Every height between 10 and 75 occurred at least once, so all of them are in the range, and each is listed once. On the water park graph, where each line passes each height exactly once, that distinction never came up. Here it does.

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Domain and Range · bookSHelf Integrated Math 1§1.4

The situation

Find the vertical intercept, the horizontal intercept, the maximum, and the minimum of f(x)=2x+6f(x) = 2x + 6.

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Domain and Range · bookSHelf Integrated Math 1§1.4

Worked example — Step

Step 1 — vertical intercept. Evaluate at 0:

f(0)=2(0)+6=6f(0) = 2(0) + 6 = 6

so the graph crosses the vertical axis at (0,6)(0, 6).

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Domain and Range · bookSHelf Integrated Math 1§1.4

Worked example — Step

Step 2 — horizontal intercept. Solve f(x)=0f(x) = 0:

2x+6=0⟹2x=−6⟹x=−32x + 6 = 0 \quad\Longrightarrow\quad 2x = -6 \quad\Longrightarrow\quad x = -3

Check it: f(−3)=2(−3)+6=0f(-3) = 2(-3) + 6 = 0. The graph crosses the horizontal axis at (−3,0)(-3, 0).

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Domain and Range · bookSHelf Integrated Math 1§1.4

Worked example — Step

Step 3 — maximum and minimum. It has neither. The line climbs forever to the right and falls forever to the left, so there is no greatest output and no least one.

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Domain and Range · bookSHelf Integrated Math 1§1.4

Worked example — Answer

Answer: vertical intercept (0,6)(0, 6), horizontal intercept (−3,0)(-3, 0), no maximum and no minimum.

A function is not required to have a maximum or a minimum, and a straight line with a nonzero slope never has either.

Take care not to swap the two intercepts. The vertical intercept is (0,6)(0, 6) — input 0, output 6. The horizontal intercept is (−3,0)(-3, 0) — input −3-3, output 0. Each one has a zero in it, but in a different slot, and mixing them up turns every answer inside out.

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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.

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