Quarry School

Read function notation as an instruction

Explain it like I am five

A recipe has a name, an ingredient amount, and a finished result. Function notation keeps those three jobs visible. In f(x), f names the recipe, x is the ingredient you supply, and f(x) names the result. Read it as 'f of x.' The parentheses mark the input slot. They do not tell you to multiply f by x. If you see f(3), the input is already chosen and you find its result. If you see f(x) = 3, the result is chosen and you find which inputs could produce it. A letter, a number, or a whole expression can fill the input slot.

xff(x)inputoutput
The name of the rule and the name of its output have different jobs.
Reminder
  • Distribution. 2(a + 2) = 2a + 4 because the multiplier reaches both terms.
Why it works. Naming the function separately from its output lets you use one rule at many inputs. The name f stays the same in f(2), f(5), and f(a + h), while the chosen input changes. Parentheses are a convention for showing that choice clearly. Treating them as multiplication would erase the instruction to apply a rule, and it would make a word input such as March meaningless as a product.
Ruley = f(x): f is the function name, x is the input, and y or f(x) is the output. Evaluate means find an output. Solve means find the input or inputs producing a specified output.
The same idea, five ways
Say it

f of two equals six.

Write it

The output of f at input 2 is 6.

In math
  • f(2) = 6
  • (2, 6)
  • When x = 2, y = 6.
  • The column with input 2 over output 6 gives the same pair.
Like

Button 2 gives the result labeled 6.

See it
2f6inputoutput
For f(x) = x2 + 3x − 4, input 2 gives output 6.
The same idea, other ways
As a recipe

f is the recipe name, x is your ingredient amount, and f(x) is the finished amount.

As three labels

In d(March) = 31, d names the rule, March names the input, and 31 is the output.

Marchdays in month31inputoutput
A word input can determine a number output.
As opposite requests

f(3) starts at input 3. f(x) = 3 starts at output 3. Put your finger on the given side before moving.

Why the letters stay useful

Once you name a rule f, you can ask for many outputs without rewriting its full recipe each time.

For f(x) = x2 + 3x − 4It asksFirst move and result
f(2)Input 2 is given. Find its output.(2)2 + 3(2) − 4 = 6.
f(a + h)The whole input a + h is given. Find its output expression.(a + h)2 + 3(a + h) − 4. Here a is the starting input and h is a step added to it. The evaluating lesson expands this expression.
f(x) = 6Output 6 is given. Find every input.Set x2 + 3x − 4 = 6. The solving lesson shows why the inputs are 2 and −5.
.1Names, numbers and units

The input does not have to be x, and the output does not have to be y. Think of labels on measuring cups. Choose letters that make the quantities recognizable. If a is a pig’s age in days, W(a) can mean its weight in pounds at that age. The definition supplies the units. The parentheses alone cannot tell you them.

  • The independent variable a measures age in days; the dependent variable W(a) measures weight in pounds.
  • h(a) can mean height at age a, if you first define that meaning.
  • Science books sometimes write y(x), using y as the rule name. This course writes y = f(x) so the rule name and the output remain separate.
  • An algebraic expression such as a + 2 combines numbers, letters and operations.
2018P240inputoutput
Input 2018 is a year. Output 240 is a count of officers.
The same idea, five ways
Say it

P of twenty eighteen equals two hundred forty.

Write it

In the input year 2018, the officer-count output is 240.

In math
  • P(2018) = 240
  • (2018, 240)
  • t = 2018 gives P(t) = 240
Like

The date label on a record tells you when its count applies.

See it
2018P240inputoutput
Input 2018 is a year. Output 240 is a count of officers.
Worked exampleExplain a statement with units

The function P(t) gives the number of officers in a town in year t. Interpret P(2018) = 240.

2018P240inputoutput
Input 2018 is a year. Output 240 is a count of officers.
What it asks. Turn the function statement into a sentence that keeps year and officer count in their correct roles.
Plan. Read the definition for units, then read input inside parentheses and output after equals.
  1. Read the input inside the parentheses: t = 2018.The model defines t as a calendar year, not an officer count.
  2. Read the output after the equals sign: 240 officers.P(t) is defined as the count of officers.
  3. State the relationship in words: in 2018 the town had 240 officers.Including both units prevents swapping the input and output.
Answer
In 2018, the town had 240 officers.
Check Write the pair (2018, 240). Its first value is the year and its second is the count, matching the interpretation.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: P(2018) = 240 means year 240 had 2018 officers.
The input and output positions were swapped.
✓ Instead: It means 240 officers in the input year 2018.
Tips and tricks
  • Read the definition of the function before interpreting its notation.
  • As labeled containers: A year and an officer count can both be numbers; their units tell you which container each belongs in.
  • As a sentence: 'Weight is a function of age' becomes W(a), after you define pounds and days.
.2Expression inputs

An expression can be an input even when you cannot turn it into a number yet. Treat a + 2 as one sealed package. Put that whole package wherever the old input variable occurred, then open it using distribution. Do not add 2 to the old output unless the formula itself tells you those results match.

  • Replace every occurrence of the input variable.
  • Parentheses protect sums and negative inputs.
  • In general j(a + 2) differs from j(a) + 2; the first changes the input, the second changes an output.
a + 2j(x) = 2x + 12a + 5inputoutput
The whole expression goes into the input slot.
The same idea, five ways
Say it

j of a plus two.

Write it

Apply the j rule to the complete input a + 2.

In math
  • j(x) = 2x + 1
  • j(a + 2) = 2(a + 2) + 1
  • j(a + 2) = 2a + 5
Like

Keep the whole ingredient package together before the recipe doubles it.

See it
a + 2j(x) = 2x + 12a + 5inputoutput
The whole expression goes into the input slot.
Worked exampleA whole expression can occupy the input slot

For j(x) = 2x + 1, evaluate j(3) and j(a + 2). Evaluate means find the output for the stated input.

a + 2j(x) = 2x + 12a + 5inputoutput
The whole expression goes into the input slot.
What it asks. Find one numerical output and one expression output for the same rule j.
Plan. Copy the entire input into the x slot. Multiply the complete parentheses by 2, then add 1.
  1. Write j(3) = 2(3) + 1 = 6 + 1 = 7.The input 3 replaces x in the rule.
  2. Write j(a + 2) = 2(a + 2) + 1.The entire expression a + 2 replaces the input variable.
  3. Distribute: 2a + 4 + 1 = 2a + 5.Two copies of a + 2 contain 2a plus 4.
Answer
  • j(3) = 7.
  • j(a + 2) = 2a + 5.
Check At a = 1 the new input is 3. The symbolic answer gives 2(1) + 5 = 7, matching j(3).
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: For j(x) = 2x + 1, j(a + 2) = j(a) + 2.
The extra 2 in the input is doubled by the rule.
✓ Instead: j(a + 2) = 2a + 5, while j(a) + 2 = 2a + 3.
Tips and tricks
  • Replace first, simplify second.
  • As a sealed package: Copy (a + 2) into every input slot before doing any arithmetic.
  • With numbers: For j(x) = 2x + 1, j(3) = 7, while j(1) + 2 = 5. Input changes and output changes differ.
.3Read and check a table of values

Think of a table as a row of paired recipe cards. Each column keeps a starting value above its result. You read down one column to keep that pair together. A table of values can define a whole small function, such as the twelve months of a nonleap year. It can also show only a few observations from a larger situation. You cannot assume it tells you missing values. To check whether it is a function, look for repeated starting values and compare their results. A repeated starting value is harmless if its result agrees; it causes a problem if two different results are assigned.

  • Columns carry the same information as ordered pairs, so the function definition does not change when the layout changes. Two columns with the same input and different outputs still make an output request ambiguous. Two columns sharing an output do not. A sample table also gives no automatic rule between its entries: many possible functions can agree at the shown inputs and disagree elsewhere.
  • A table is a function when no input appears with two different outputs. Outputs may repeat. A table listing every accepted input gives the entire domain; a table of selected values gives only the information shown.
  • Read the input and output labels and units.
  • Keep each column together as one ordered pair.
  • Locate every repeated input and compare all its outputs.
  • Accept matching repeats and shared outputs; reject conflicting outputs.
  • Decide whether the definition is complete or only a sample before answering about missing inputs.
input xoutput g(x)−350145
The output 5 repeats, while each input has one answer.
Reminder
  • Function definition. One input must settle one output. Shared outputs, such as −3 → 5 and 4 → 5, are allowed.
The same idea, five ways
Say it

The column under two says f of two equals one.

Write it

A table column is one input-output pair.

In math
  • f(2) = 1
  • (2, 1)
  • Input 2 → output 1
  • The output directly below input 2 is 1.
Like

Read the two halves of one paired recipe card together.

See it
input xoutput f(x)215386
Each listed input has one output.
Worked exampleReading function notation from a table: a cooling cup of tea

A cup of tea is poured and left on a desk. The function T gives the tea's temperature, T(m), in degrees Celsius, m minutes after pouring. The table lists every value of T that we know:

m (minutes): 0, 5, 10, 15, 20
T(m) (°C): 90, 74, 62, 54, 48

(a) Evaluate T(10) and say what it means.
(b) Solve T(m) = 54 and say what it means.
(c) A classmate says T(5 + 15) = T(5) + T(15) = 74 + 54 = 128 °C. Find T(5 + 15) correctly and explain the mistake.

input time m (minutes)output temperature T(m) (°C)090574106215542048↓ evaluate: input given, read the output below it
Table of the tea's temperature T(m) in °C at m = 0, 5, 10, 15, 20 minutes (90, 74, 62, 54, 48), with the inputs 10 and 20 highlighted for evaluation.
  1. Name the parts: the function is T, the input m is the time in minutes since pouring, and the output T(m) is the temperature in °C.Before reading any value, we need to know which row of the table is the input and which is the output.
  2. (a) Find the input 10 in the m row. The entry below it in the T(m) row is 62, so T(10) = 62.Evaluating means finding an output. We start from the given input and read across to its output.
  3. Interpret: T(10) = 62 means that 10 minutes after pouring, the tea is 62 °C.An answer in function notation should be stated with its units and in terms of the situation.
  4. (b) Find 54 in the T(m) row. The input above it is m = 15, so the solution is m = 15.Solving means finding the input that gives a specified output. Here we start from the output 54 and read back to the input.
  5. Check the T(m) row for any other entry equal to 54. There is none, so m = 15 is the only solution in the table. Interpret: the tea is 54 °C exactly 15 minutes after pouring.Solving asks for every input that gives the output, so we must not stop at the first match without checking the rest.
  6. (c) Treat everything inside the parentheses as one input: 5 + 15 = 20. Then T(5 + 15) = T(20) = 48 °C.The parentheses hold a single input. We simplify it first, then apply the function once.
  7. Explain the error: the classmate applied T separately to 5 and to 15 and added the outputs. T(5) + T(15) = 128 °C is the sum of two temperatures and does not describe the tea at 20 minutes.T(a + b) is one instruction with a single input. It is not the same as T(a) + T(b), which applies T twice.
Answer
(a) T(10) = 62: ten minutes after pouring, the tea is 62 °C. (b) m = 15: the tea is 54 °C at 15 minutes. (c) T(5 + 15) = T(20) = 48 °C. The classmate wrongly split T across the sum.
Check Each result can be read back from the table. In the column m = 10 the temperature is 62. The temperature 54 appears only once, in the column m = 15. In the column m = 20 the temperature is 48. The temperatures fall steadily (90, 74, 62, 54, 48), so a value of 128 °C at 20 minutes is impossible. It is even higher than the starting temperature of 90 °C, which confirms the classmate's answer is wrong.

Work to write

  1. T(10) = 62, so after 10 minutes the tea is 62 °C
  2. T(m) = 54 when m = 15, so the tea is 54 °C after 15 minutes
  3. 5 + 15 = 20, so T(5 + 15) = T(20) = 48 °C
  4. T(5 + 15) ≠ T(5) + T(15), because the whole sum is a single input

(a) T(10) = 62: ten minutes after pouring, the tea is 62 °C. (b) m = 15: the tea is 54 °C at 15 minutes. (c) T(5 + 15) = T(20) = 48 °C. The classmate wrongly split T across the sum.

Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: The second table is not a function because its output 5 repeats.
A function permits different inputs to have the same answer.
✓ Instead: It is a function because each input has exactly one output.
✗ Not this: Three listed pairs determine every value between them.
Different unseen rules can share the same three shown pairs.
✓ Instead: Say the missing value is not determined by this table alone.
Tips and tricks
  • A repeated identical pair such as (2, 4) listed twice does not create a second output.
  • Never connect table dots or guess values in between unless a formula tells you those extra values.
  • For a yes-or-no answer, write the input and the conflicting outputs, such as not a function: input 5 has outputs 2 and 4.
  • A repeated output alone is allowed. Do not confuse the input row with the output row.
  • As paired cards: A column is one record, so reading across unrelated columns breaks the pairing.
  • As a list of arrows: The columns of the second table mean −3 → 5, 0 → 1, and 4 → 5. Two arrows arriving at 5 are allowed.
.4A complete month table

This table uses month numbers rather than month names. January is 1, February is 2, and so on through December at 12. That change labels the inputs differently without changing the calendar counts. The nonleap-year condition fixes February at 28 days. A nonleap year has 28 days in February. A leap year has 29.

  • Every one of the twelve accepted inputs is shown.
  • Repeated day counts are allowed.
  • The domain is a finite set of month integers, not every number between 1 and 12.
input moutput d(m)131228331430531630
January through June: m is month number and d(m) is days in a nonleap year.
The same idea, five ways
Say it

d of three equals thirty-one.

Write it

Month input 3, March, gives output 31 days.

In math
  • d(3) = 31
  • (3, 31)
  • Domain: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
  • Range: {28, 30, 31}
Like

Month numbers are labels on twelve calendar pages, not all numbers between 1 and 12.

See it
input moutput d(m)131228331430531630
January through June: m is month number and d(m) is days in a nonleap year.
Worked exampleReading a complete month table: a bakery's loaves sold

A bakery records how many loaves of bread it sells in each month of one year. The function B gives the number of loaves sold, B(n), in hundreds of loaves, during month n, where n = 1 is January and n = 12 is December. The table lists every value of B for that year:

n (month): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
B(n) (hundreds of loaves): 42, 38, 45, 51, 56, 60, 63, 60, 52, 47, 49, 70

(a) Evaluate B(5) and say what it means.
(b) Evaluate B(2·3 + 1) and say what it means.
(c) Solve B(n) = 60 and say what it means.
(d) Solve B(n) = 40.

input month noutput loaves sold (hundreds)142238345451556660763860952↓ evaluate: input given, read the output below it
Monthly bread sales B(n), in hundreds of loaves, for months n = 1 (January) through 12 (December). The columns for the evaluated inputs n = 5 and n = 7 are highlighted.
  1. Name the parts. B is the function name. The input n is a month number from 1 to 12. The output B(n) is the number of loaves sold in that month, in hundreds of loaves.Before reading the table, we need to know which row is the input and which is the output, and what units the output carries.
  2. (a) The input is 5. Find n = 5 in the input row and read the output below it: B(5) = 56.To evaluate is to find an output. For a function given by a table, the rule is to look up the input and read its output.
  3. Attach meaning and units: 56 hundred loaves is 5600 loaves. In May (month 5) the bakery sold 5600 loaves.The output is measured in hundreds of loaves, so an answer with no units would not describe the sales.
  4. (b) Treat everything inside the parentheses as one input and simplify it first: 2·3 + 1 = 7. So B(2·3 + 1) = B(7).Whatever sits inside the parentheses is the whole input. B(2·3 + 1) does not mean B(2)·3 + 1.
  5. Read the table at n = 7: B(7) = 63. In July the bakery sold 63 hundred loaves, which is 6300 loaves.Once the input has been simplified to a single month number, we evaluate it exactly as in part (a).
  6. (c) Search the output row for every entry equal to 60. It appears at n = 6 and at n = 8, so n = 6 or n = 8.To solve is to find the inputs that give a stated output. We scan the whole output row, because more than one input can give the same output.
  7. Interpret the result: the bakery sold 6000 loaves in June and also in August, and in no other month.The table lists every month, so there are no other solutions.
  8. (d) Search the output row for 40. No entry equals 40. The smallest value is 38, in February, and 42 is the next value above 40. So B(n) = 40 has no solution.A solve question can have no answer. This table covers the whole year, so no month had sales of exactly 4000 loaves.
Answer
(a) B(5) = 56: in May the bakery sold 56 hundred loaves (5600 loaves). (b) B(2·3 + 1) = B(7) = 63: in July it sold 63 hundred loaves (6300 loaves). (c) n = 6 or n = 8: sales were 6000 loaves in June and in August. (d) No solution: in no month were exactly 4000 loaves sold.
Check Substitute each result back into the table. The entry under n = 5 is 56 and the entry under n = 7 is 63. The entries under n = 6 and n = 8 are both 60, and every other entry differs from 60. Going through all twelve outputs (42, 38, 45, 51, 56, 60, 63, 60, 52, 47, 49, 70) shows that none equals 40.

Work to write

  1. B is the function; n = month number (input); B(n) = loaves sold in hundreds (output)
  2. B(5) = 56, so 5600 loaves were sold in May
  3. 2·3 + 1 = 7, so B(2·3 + 1) = B(7) = 63, which is 6300 loaves in July
  4. B(n) = 60 when n = 6 or n = 8 (June and August)
  5. B(n) = 40 has no solution, because no table entry equals 40

(a) B(5) = 56: in May the bakery sold 56 hundred loaves (5600 loaves). (b) B(2·3 + 1) = B(7) = 63: in July it sold 63 hundred loaves (6300 loaves). (c) n = 6 or n = 8: sales were 6000 loaves in June and in August. (d) No solution: in no month were exactly 4000 loaves sold.

Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: The table accepts month 3.5 because 3.5 lies between 3 and 4.
Month numbers are integer labels; this domain does not include fractional months.
✓ Instead: Use one of the twelve listed integer inputs.
✗ Not this: Write the month range as [28, 31].
An interval includes every real number between its endpoints, while the table produces only three separate day counts.
✓ Instead: Range: {28, 30, 31}.
Tips and tricks
  • State whether the year is a leap year before fixing February's output.
  • Use braces for the separate day counts {28, 30, 31}. The interval [28, 31] would also include 29 and 30.5, which no month in this table has.
  • As a calendar: Several months last 31 days, but each named month has one length in this year.
  • As relabeling: input March and input 3 identify the same month under two different definitions.
  • As relabeling: Input March and input 3 identify the same month under two different definitions.
.5Conflicting age data

A group of children can contain two children of the same age and different heights. That is useful data, but age alone cannot predict a unique listed height for that group. The question is about the defined input, not about whether the measurements are sensible.

  • Age 5 occurs with both 40 inches and 42 inches.
  • Those two outputs make the relation fail to be a function of age for this group.
input ageoutput inches5405426447478509521054↓ evaluate: input given, read the output below it
The two age-5 columns have different heights.
The same idea, five ways
Say it

Age five gives forty inches and forty-two inches.

Write it

One input, age 5, has two different listed height outputs.

In math
  • (5, 40)
  • (5, 42)
  • 5 → 40 and 5 → 42
Like

Two five-year-old children need not have the same height.

See it
input ageoutput inches5405426447478509521054↓ evaluate: input given, read the output below it
The two age-5 columns have different heights.
Worked exampleA repeated input can conflict

Does the age-height table make height a function of age for the whole listed group?

input ageoutput inches5405426447478509521054↓ evaluate: input given, read the output below it
The two age-5 columns have different heights.
What it asks. Does age alone select one height in this table?
Plan. Find all columns for age 5 and compare their heights. One conflicting input is enough to reject the relation as a function.
  1. Find both columns with input age 5.The same starting input must be checked wherever it appears.
  2. Read their outputs: 40 inches and 42 inches.Those are different heights for the same input.
  3. Classify the relation as not a function of age alone.Age 5 does not determine one height for this group.
Answer
Not a function of age alone.
Check Asking for height at age 5 gives two answers, 40 and 42. The table cannot choose one of them.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Every age value looks reasonable, so the table defines a function.
Plausible data can still assign different outputs to the same input.
✓ Instead: Test uniqueness for age 5.
Tips and tricks
  • Find duplicates in the input row before inspecting the output row.
  • As two children: Knowing that a child is five does not tell you which of two different five-year-olds is meant.
  • As two arrows: One age input pointing to 40 and 42 gives two competing answers.
.6Repeated outputs and conflicting inputs

Output repetition and input conflict look similar until you read the row labels. In the first table below, the two 5s are answers. Inputs −3 and 4 each have one answer. In the second table of the worked example, the two 5s are starting values, and their answers are 2 and 4. Those positions explain why the first table is a function and the second table is not.

  • The first table is a function because no input has competing answers.
  • The second table is not a function because input 5 has outputs 2 and 4.
  • Only give a table a function name after it passes the function check.
input xoutput g(x)−350145↑ solve: output given, read every input above it
The output 5 repeats, while each input has one answer.
The same idea, five ways
Say it

Same output is allowed. Same input with different outputs is not.

Write it

Different inputs can share one output; one input cannot choose between two outputs.

In math
  • (−3, 5) and (4, 5): allowed
  • (5, 2) and (5, 4): not a function
Like

Two buttons may give the same snack, but one button must not have two assigned snacks.

See it
input Inputoutput Output105254↓ evaluate: input given, read the output below it
Input 5 has two different outputs.
Worked exampleWhy one repeated 5 is allowed and the other is not

Use the first and second table pictures below. Decide which table is a function.

input xoutput g(x)−350145↑ solve: output given, read every input above it
The output 5 repeats, while each input has one answer.
input Inputoutput Output105254↓ evaluate: input given, read the output below it
Input 5 has two different outputs.
What it asks. Compare a repeated output with a repeated input.
Plan. Point to the input row in each picture, then compare the columns carrying the repeated number.
  1. In the first table, inputs −3 and 4 both give 5, while input 0 gives 1.Every input still has one assigned output.
  2. In the second table, input 5 gives 2 in one column and 4 in another.The same input has two different outputs.
Answer
  • First table: function.
  • Second table: not a function. Input 5 gives 2 and 4.
Check Read down from any first-table input and you get one answer. Read down from 5 in the second table and there are two competing answers.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Reject the first table because output 5 appears twice.
The repeated 5 is an output. Inputs −3 and 4 each have one output, so that repetition is allowed.
✓ Instead: Accept the first table. Reject the second because input 5 gives both 2 and 4.
Tips and tricks
  • Point to the row labels before interpreting a repeated number.
  • By position: A repeated number matters differently in the input row and the output row.
  • By a request: Ask for the output at input 5 in the second table; there is no unique answer.
Strategy: step by step
  1. Identify the function name and what its input and output represent.
  2. Read everything inside the parentheses as the entire input.
  3. For evaluate, apply the rule to that input.
  4. For solve, require the rule's output to equal the given value.
  5. Include units when the quantities have units.
Strategy
Choose evaluation or solving
1
Is the input already given inside the parentheses, as in f(3)?
YesEvaluate: apply the rule to the whole given input and find its output.
NoRead which quantity is given.
↓
2
Is the output given after the equals sign, as in f(x) = 3?
YesSolve: set the formula equal to that output and find every allowed input. The solving lesson works this method.
NoUse the definition to identify the input, output, and units before calculating.
  1. Identify the function name and what its input and output represent.
  2. Read everything inside the parentheses as the entire input.
  3. For evaluate, apply the rule to that input.
  4. For solve, require the rule's output to equal the given value.
  5. Include units when the quantities have units.
Worked exampleEvaluating and solving with a rental-cost function

A kayak rental shop charges according to C(h) = 15 + 9h, where h is the number of hours rented and C(h) is the total cost in dollars.
(a) Evaluate C(4) and say what it means.
(b) Solve C(h) = 69 and say what it means.
(c) Write C(h + 1) as an expression in h.

input hours rented, houtput cost C(h) ($)233451669↓ evaluate: input given, read the output below it
Values of C(h) = 15 + 9h at h = 2, 4 and 6 hours: $33, $51 and $69. The input h = 4 is marked as the one evaluated in part (a), and the output $69 at h = 6 is the solution to part (b).
  1. Name the parts: the function is C, the input h is time in hours, and the output C(h) is cost in dollars.In y = f(x), the letter outside the parentheses names the function, the letter inside is the input, and f(x) is the output. Units come from what each quantity measures.
  2. (a) Substitute 4 for h: C(4) = 15 + 9(4) = 15 + 36 = 51.Evaluate means find the output. The number inside the parentheses is the input, so h = 4 is fed into the rule.
  3. (a) Interpret: C(4) = 51 dollars, so renting for 4 hours costs $51.The output carries the output's units, which are dollars.
  4. (b) Set the rule equal to the given output: 15 + 9h = 69.Solve means find the input that produces a given output. Here the output C(h) is 69, so the rule must equal 69.
  5. (b) Subtract 15 from both sides: 9h = 54. Then divide by 9: h = 6.Undoing the operations in reverse order isolates the input.
  6. (b) Interpret: h = 6 hours, so a $69 bill means the kayak was rented for 6 hours.The answer to a solve question is an input, so it carries the input's units, which are hours.
  7. (c) Replace every h in the rule with the whole input (h + 1): C(h + 1) = 15 + 9(h + 1).Everything inside the parentheses is the entire input. Writing (h + 1) in place of h keeps that input together.
  8. (c) Simplify: 15 + 9h + 9 = 9h + 24.Distribute the 9 to both terms of the input, then combine the constants.
Answer
(a) C(4) = 51: a 4-hour rental costs $51. (b) h = 6: a $69 rental lasted 6 hours. (c) C(h + 1) = 9h + 24 dollars, the cost of renting one hour longer than h hours.
Check (a) 9 × 4 = 36 and 36 + 15 = 51. (b) Substituting h = 6 gives C(6) = 15 + 54 = 69, which matches the given output. (c) Test with h = 3. Then C(h + 1) = C(4) = 51, and 9(3) + 24 = 27 + 24 = 51. Both agree.

Work to write

  1. C is the function, h = hours rented (input), C(h) = cost in dollars (output)
  2. C(4) = 15 + 9(4) = 51
  3. C(4) = 51 dollars: 4 hours costs $51
  4. 15 + 9h = 69
  5. 9h = 54
  6. h = 6 hours
  7. C(h + 1) = 15 + 9(h + 1) = 9h + 24

(a) C(4) = 51: a 4-hour rental costs $51. (b) h = 6: a $69 rental lasted 6 hours. (c) C(h + 1) = 9h + 24 dollars, the cost of renting one hour longer than h hours.

Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: f(3) = 3f.
f is a rule name, and the parentheses select input 3.
✓ Instead: Apply the defined rule at input 3.
✗ Not this: Solve f(x) = 3 by plugging 3 in for x.
The 3 is the requested output, not the known input.
✓ Instead: Set the formula equal to 3 and find all allowed inputs.
✗ Not this: f(x + 1) = fx + f, by distribution.
Here f names a function, not a multiplying number. Only the defined formula tells you the output.
✓ Instead: For the rule f(x) = 2x + 1, f(x + 1) = 2(x + 1) + 1 = 2x + 3.
Tips and tricks
  • Memory cue: inside names the input; equals names the output.
  • Use descriptive letters for word problems, then define their units.
  • A letter in front of parentheses is a function name when a definition names it as a rule, such as f(x) = 2x + 1. A number in front multiplies: 3(x + 1) = 3x + 3. In a difference quotient, h names a step number, so h(2a + h + 3) means multiplication.
  • Function names are name tags: f, g, h, q, r, s, d, M, P, and W can all name rules. Read the formula or table that defines the chosen name.
Trap. Reading f(x) as f times x. A function name identifies a rule rather than a number being multiplied.