Algebra 1 · F.IF.B.5

A reasonable domain for a real situation

A formula will take any number you give it. When the formula describes something real, which of those numbers still make sense?

You need

Domain and range from a graphAlgebra 1 Function notation in a real situationAlgebra 1

Unlocks

Where a model holds: domain, interpolation and extrapolationAlgebra 1

Where you are 4 in Domain and range

Functions13

What a function is5

Domain and range4

Domain and range from pairs and tables Domain and range from a graph

A reasonable domain for a real situation

The implied domain of a formula

Why this exists

A formula is happy with any number. Feed P(n) = 14n - 96 the input n = -2.5 and it hands back -131 without complaint. But if n is the number of books a shop has sold, -2.5 is nonsense: nobody sells minus two and a half books.

So a function that describes a real situation carries, beside its rule, the set of inputs the situation allows. That set decides whether the graph is dots or an unbroken line, where it stops, which outputs can happen, and which solutions of an equation to keep.

Earlier lessons met this idea in passing: reading a statement such as C(12) = 40 asked which inputs make sense, graphing from a table asked whether to join the dots, building a linear model asked where a model stops, and reading a set of pairs collected a domain that was handed over. This lesson asks where the domain comes from.

Definition

The domain of a function is the set of inputs it accepts, and the range is the set of outputs it gives (see Domain and range). When a function describes a real situation, its reasonable domain is the set of inputs that make sense in that situation. The range then follows from it: it is the set of outputs the rule gives for inputs in the reasonable domain.

The limits on a reasonable domain come from four places:

What the input counts or measures. A count of people, books or cars takes only the whole numbers 0, 1, 2, 3 and so on. A time, a length or a mass can take any value in between.
Where the story starts. Time measured from the moment a hose is turned on cannot be negative.
A limit in the world or in the rules. A shop has only so many copies; a ride lasts a set number of minutes; a pool holds only so much.
A limit on the output, carried back to the input. A height cannot fall below 0 and a pool cannot hold more than it holds, so the inputs that would give such outputs are left out.

A function is discrete when its domain is a set of separate values with gaps between them, such as \{0, 1, 2, \ldots, 12\}. Its graph is a set of separate dots. The values need not be whole numbers: fabric sold in half-meter lengths has the separate inputs 0.5, 1, 1.5, \ldots

A function is continuous when its domain is an unbroken stretch of numbers and the output changes without sudden jumps, so its graph can be drawn without lifting the pen. Its graph is an unbroken line or curve, stopping where the domain stops.

A reasonable domain can be written four ways:

Form	Minutes of filling	Copies sold
Words	every time from 0 to 34	whole numbers 0 to 12
Inequality	0 \le t \le 34	0 \le n \le 12, n whole
Interval	[0, 34]	none
List	none	\{0, 1, \ldots, 12\}

In interval notation a square bracket means the end is included and a round bracket, as in (0, 5], means it is left out. An interval means every number in the stretch, so it never describes a count, and no list can hold every number in a stretch.

Intuition

Ask of the input: could it be halfway between two of its values? Halfway between 3 and 4 minutes is 3.5 minutes, and the water in a pool does hold some amount at that moment, so the graph is joined. Halfway between 3 and 4 books sold is nothing at all, so the graph is dots. What decides is the input, not the rule: Figure 4 draws one rule both ways. Then ask where the story begins and ends. The graph covers only those inputs, and the outputs it reaches are the range.

Results

Every limit at once. The reasonable domain is the set of inputs that satisfy every limit together. When there are several upper limits, the smallest one decides; when there are several lower limits, the largest one decides. Writing each limit as an inequality is the work of Writing and interpreting constraints.

The range of a straight-line rule on an interval. If f(x) = mx + c and the domain is a \le x \le b, the range runs between the two end outputs:

\begin{aligned} m > 0:&\quad f(a) \le f(x) \le f(b) \\ m < 0:&\quad f(b) \le f(x) \le f(a) \end{aligned}

An end left out of the domain is left out of the range too.

The range of a rule on a list. Apply the rule to every input in the list. The outputs, and only those, form the range.

Why it is true

The smallest upper limit decides. Suppose an input must satisfy t \le 34 and also t \le 40. Any t with t \le 34 already has t \le 40, so the second limit removes nothing. Turned round, the same argument shows the largest lower limit decides.

The range of a straight line. Take m > 0 and a \le x \le b. Multiplying by the positive number m keeps the inequalities pointing the same way, and adding c to every part keeps them too:

\begin{aligned} ma + c &\le mx + c \le mb + c \\ f(a) &\le f(x) \le f(b) \end{aligned}

So no output lies outside that stretch. Every output inside it is reached as well: for any y between f(a) and f(b), the input \displaystyle x = \frac{y - c}{m} gives f(x) = y, and undoing the steps above shows a \le x \le b. When m < 0, multiplying by m reverses both inequalities, which is why the ends swap. A strict end of the domain stays strict through the same steps.

Worked examples

Example 1

A pool filling from a hose

Marigold fills a paddling pool with a garden hose. The pool already holds 90 liters of rainwater and holds 600 liters when full. The hose adds 15 liters a minute, so t minutes after she turns it on the pool holds V(t) = 90 + 15t liters. She turns it off when the pool is full. Give the reasonable domain and range in words, as inequalities and in interval notation, and say how the graph looks.

Solution

Time is measured from when the hose goes on, so t \ge 0. The pool is full when

\begin{aligned} 90 + 15t &= 600 \\ 15t &= 510 \\ t &= 34 \end{aligned}

after which the formula would describe water the pool cannot hold. So the domain is every time from 0 to 34 minutes: 0 \le t \le 34, or [0, 34].

Time passes through every value in between, and the water rises without jumps, so the function is continuous and the graph is a joined segment from t = 0 to t = 34 (Figure 1).

Figure 1The pool of Example 1. The segment starts at 90 liters and stops when the pool is full, at 34 minutes.

The rule rises with t, so the range runs from V(0) = 90 to V(34) = 600: every amount from 90 to 600 liters, 90 \le V \le 600, or [90, 600]. The range does not start at 0, because the pool was never empty.

Example 2

Profit on a stock of books

A bookshop orders 12 signed copies of a new novel, paying 96 dollars for the lot, and sells them at 14 dollars each. After n copies are sold, its profit on the order is P(n) = 14n - 96 dollars. Give the reasonable domain and range, and say how the graph looks.

Solution

n counts copies, so it is a whole number, and the shop cannot sell more than the 12 it has. The domain is the list \{0, 1, 2, \ldots, 12\}. The function is discrete.

The range is the list of outputs. P(0) = -96, and each extra copy adds 14:

\begin{aligned} \{&-96, -82, -68, -54, -40, -26, -12, \\ &2, 16, 30, 44, 58, 72\} \end{aligned}

The last is P(12) = 168 - 96 = 72. The range includes negative numbers, and rightly: a negative profit is a loss, and until enough copies sell the shop is out of pocket. A limit on the output comes from the situation, never from a habit that outputs must be positive.

The graph is 13 separate dots (Figure 2). They lie on a straight line, but joining them would claim the shop can sell 2.5 copies.

Figure 2The bookshop of Example 2. Copies come whole, so the graph is thirteen separate dots, and it stops at the twelfth copy.

Example 3

Discrete or continuous

For each function, say whether it is discrete or continuous and give a reasonable domain.

(a) The fare income of a river trip by the number of passengers, on a boat that seats 24.

(b) The height of a loaf as it rises, by the minutes since it was set to rise, for a 90-minute rise.

Solution

(a) Passengers come whole, and there are at most 24. Discrete, with domain \{0, 1, 2, \ldots, 24\}, where 0 passengers gives no income.

(b) Time passes through every value, and the loaf grows without jumps. Continuous, with domain 0 \le t \le 90, or [0, 90].

(c) The possible lengths are 0.5, 1, 1.5, \ldots, 4 meters, with nothing allowed in between. Discrete, with domain \{0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4\}: eight values, none of them a count. Discrete means separate values, not whole numbers.

Example 4

A sandcastle and the turning tide

Quincy builds a sandcastle 36 cm tall. Once the first wave reaches it, each wave washes some away, and its height falls at a steady 1.5 cm a minute, so t minutes after the first wave its height is H(t) = 36 - 1.5t cm. The tide turns 20 minutes after the first wave reaches the castle, and after that the water no longer gets to it. Give the reasonable domain and range.

Solution

Time is measured from the first wave, so t \ge 0. There are two upper limits. The castle would be flat when

36 - 1.5t = 0 \quad\Longrightarrow\quad t = 24

and the waves stop reaching it at t = 20. The smaller limit decides, since after t = 20 the castle stops shrinking and the rule no longer describes it. The domain is 0 \le t \le 20, or [0, 20].

The rule falls as t grows, so the ends of the range swap: it runs from H(20) = 36 - 30 = 6 up to H(0) = 36. The range is 6 \le H \le 36, or [6, 36].

Figure 3The sandcastle of Example 4. The solid part is the reasonable domain; the dashed part is the formula carrying on after the tide has turned.

The formula says H(24) = 0, but that is a fact about the formula, not about the castle. In the situation a 6 cm stump survives the tide (Figure 3).

Example 5

When does the order make a profit?

For the bookshop of Example 2, for which numbers of copies sold is the order in profit, and what profits are possible then? Can the profit ever be exactly 50 dollars?

Solution

In profit means P(n) > 0. Solve the inequality as usual (see Solving linear inequalities):

\begin{aligned} 14n - 96 &> 0 \\ 14n &> 96 \\ n &> \tfrac{48}{7} \approx 6.86 \end{aligned}

The algebra alone would allow n = 6.9, but the domain allows only whole numbers up to 12. The whole numbers above 6.86 in the domain are n = 7, 8, 9, 10, 11, 12. After 6 copies the shop is still 12 dollars down; the seventh copy turns that into a profit of 2 dollars. The possible profits are \{2, 16, 30, 44, 58, 72\}.

For exactly 50 dollars, solve 14n - 96 = 50: then 14n = 146 and \displaystyle n = \frac{73}{7}, about 10.43. That is not a whole number, so it is not in the domain, and the profit is never exactly 50 dollars. For the same reason the shop never exactly breaks even.

Example 6

One rule, two domains

A market stall sells loose cherries at 3 dollars a kilogram, plus 2 dollars for a basket that holds up to 5 kg. It also sells honey at 3 dollars a jar, plus 2 dollars for a gift box that holds up to 5 jars. Both costs follow C = 3x + 2. Give the reasonable domain and range of each.

Solution

(a) For the cherries, x is a mass, so it can be any number up to 5. A purchase has some cherries in it, so x > 0; the domain is 0 < x \le 5, or (0, 5]. The rule rises, so the range runs from just above C(0) = 2 up to C(5) = 17. Since 0 is left out of the domain, 2 is left out of the range: 2 < C \le 17, or (2, 17]. The graph is a segment with an open circle at its left end.

(b) For the honey, x counts jars, at least one and at most 5: the domain is \{1, 2, 3, 4, 5\}, and the range is \{5, 8, 11, 14, 17\}. The graph is five dots.

Figure 4One rule, C = 3x + 2, with two different domains. The open circle in (a) marks an end that is left out.

The two functions share a rule and even an upper limit, yet their graphs and ranges differ. The domain comes from the story, never from the formula.

Boundaries

A reasonable domain is judged, not computed. Every input in it must make sense; whether 0 belongs (no copies sold, no time passed) can be argued either way, so state the choice and why.

A huge count is often treated as continuous. A crowd of thousands comes in whole people, but the gaps are too small to matter, so a model may join the graph.

An unbroken domain does not promise an unbroken graph. A parking lot that charges for each hour begun accepts any time, yet its price jumps at each whole hour: a piecewise rule.

Reading the range off the ends needs a rule that only rises or only falls. A ball thrown upwards rises and then falls, and its greatest height comes in the middle of its domain, not at an end.

Inside the domain the model is still a model. A steady rate is usually an approximation, and predicting beyond the inputs the data covered is riskier still: see Where a model holds: domain, interpolation and extrapolation.

Where this goes next

Here the domain came from a story. On a graph it can be read straight off the picture, as in Domain and range from a graph, and for a bare formula with no story it is every input the formula can accept, the The implied domain of a formula of the formula. A sequence is a discrete function whose domain is the whole numbers, with its outputs listed in order. And reading what the starting value and rate of a rule mean in a situation is Linear functions in context.

Exercises

Discrete or continuous

Say whether each function is discrete or continuous, and give a reasonable domain.

The takings from hiring out rowing boats, by the number of boats hired, from a fleet of 15.
The temperature in a greenhouse, by the hours since sunrise, over the 6 hours until noon.

Domain and range from a story

A full tea urn holds 18 liters. Its tap drains it at 0.6 liters a minute, so after t minutes it holds W(t) = 18 - 0.6t liters. Give the reasonable domain and range in interval notation.
Nils washes cars for 9 dollars each and has time for at most 6 on Saturday. His takings are E(c) = 9c for c cars. Give the reasonable domain and range as lists.
A snowdrift 45 cm deep melts down 2.5 cm an hour, so after t hours it is H(t) = 45 - 2.5t cm deep. Give the reasonable domain and range if it melts until it is gone. What does H(20) give, and what does it mean?
A bookshop orders 30 copies of an atlas for 210 dollars and sells them at 12 dollars each, so its profit is P(n) = 12n - 210. Give the reasonable domain, the least and greatest possible profit, and the fewest copies that put the order in profit.
Rosalie swims a 1500-meter open-water course at 50 meters a minute. The distance she has left is D(t) = 1500 - 50t. Give the reasonable domain and range.

Writing a domain

A domain is 0 < m \le 8. Write it in interval notation and in words.
A team fields between 5 and 9 players. Explain why [5, 9] does not describe the possible numbers of players, and write the domain correctly.

Two limits, and working backwards

Pembe sells cups of fresh orange juice at 1.50 dollars each. She has 3 liters of juice, each cup takes 0.25 liters, and she has only 10 cups. Her takings are R(c) = 1.5c. Give the reasonable domain and range.
A delivery van starts with a full 54-liter tank and uses 0.09 liters per kilometer, so after d km it has F(d) = 54 - 0.09d liters. Today’s round is 400 km. Give the reasonable domain and range for today.
A sandcastle’s height falls at a steady rate until it is flat. Its height function has reasonable domain [0, 30], in minutes, and range [0, 42], in centimeters. How fast does it fall, and what is the rule for its height?

Reasoning

Explain why a reasonable domain is always part of the set of inputs the formula itself accepts, and never more.
Quincy says the dots in Figure 2 should be joined, because they lie on a straight line. Explain why they should not.
A function has domain \{0, 1, 2, \ldots, 10\} and rule f(n) = 3n + 1. Is 15 in its range? Is 31? Explain.
Describe a situation whose reasonable domain is 0 \le x \le 8, and one whose reasonable domain is \{0, 1, 2, \ldots, 8\}.

Answers

Discrete, \{0, 1, 2, \ldots, 15\}: boats are hired whole, and there are only 15.
Continuous, [0, 6]: time passes through every value, and the temperature has no jumps.
Domain [0, 30], range [0, 18]. The urn is empty when 18 - 0.6t = 0, at t = 30; the rule falls, so the range runs from W(30) = 0 to W(0) = 18.
Domain \{0, 1, 2, 3, 4, 5, 6\}, range \{0, 9, 18, 27, 36, 45, 54\}: whole cars, at most 6, each output 9 times its input.
Domain [0, 18], range [0, 45]. The snowdrift is gone when 45 - 2.5t = 0, at t = 18. H(20) = -5 means nothing: 20 hours is outside the domain, and no snowdrift is -5 cm deep.
Domain \{0, 1, 2, \ldots, 30\}; least profit -210 dollars, at n = 0; greatest 150 dollars, at n = 30; 18 copies, since 12n > 210 gives n > 17.5, and P(18) = 6 dollars.
Domain [0, 30], range [0, 1500]: she finishes when 1500 - 50t = 0, at t = 30 minutes.
(0, 8]: every number greater than 0, up to and including 8.
[5, 9] holds every number from 5 to 9, such as 6.5, and no team has six and a half players. The domain is \{5, 6, 7, 8, 9\}.
Domain \{0, 1, 2, \ldots, 10\}, range \{0, 1.5, 3, 4.5, \ldots, 15\}. The juice fills 3 \div 0.25 = 12 cups, but she has only 10, the smaller limit. R(10) = 15.
Domain [0, 400], range [18, 54]. The tank would run dry at d = 600, but the round ends at 400, the smaller limit, where F(400) = 54 - 36 = 18.
1.4 cm a minute, and H(t) = 42 - 1.4t. It starts at 42 cm, the top of the range, and loses all of it in 30 minutes: 42 \div 30 = 1.4.
The reasonable domain keeps only the inputs that make sense in the situation, and the formula must still give an output for each of them. The situation can only remove inputs, never add any.
The input counts copies sold, so only whole numbers make sense. A joined line would include points such as (2.5, -61), claiming the shop sold two and a half copies. Lying on a line shows the profit grows at a steady rate; it does not fill the gaps between counts.
15 is not; 31 is. 3n + 1 = 15 gives \displaystyle n = \frac{14}{3}, not in the domain. 3n + 1 = 31 gives n = 10, which is in the domain.
Many answers. The water in a tank, by the minutes a tap has run, when it is full after 8 minutes, has domain 0 \le x \le 8. The cost of a box of eggs, by the number of eggs in it, for a box that holds 8, has domain \{0, 1, 2, \ldots, 8\}.