Functions as rules pairing each input with exactly one output, input-output tables, the domain and the range, deciding whether a pairing is a function, and moving between the four representations of a function: words, a rule, a table and a graph.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 1 — Connections to Algebra
An Introduction to Functions
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.8 An Introduction to Functions §1.8, pp. 48-53 — the lesson these objectives are drawn from
Warm-up
You met the idea in Lesson 1.1 under the name formula and in Lesson 1.7 as a table. The word function is about to tie them together.
Discussion prompt
In Lesson 1.1 you used the rule 180t for distance. Feed it the inputs 1, 2 and 3 and write the outputs. Then answer this: could any single input ever produce two different outputs?
Hint: Try to construct such a case and see what stops you.
Answer:
\[ t = 1 \rightarrow 180 \qquad t = 2 \rightarrow 360 \qquad t = 3 \rightarrow 540 \]
No single input can produce two outputs, because the rule is a calculation and a calculation on a fixed number gives a fixed answer. That property — one input, exactly one output — turns out to be the definition of a function, and you have been relying on it since the first lesson of the book without needing a name for it.
Concept
A function is a rule that establishes a relationship between two quantities, called the input and the output. For each input there is exactly one output, even though two different inputs may give the same output.
function — A rule pairing each input with exactly one output. Two different inputs may share an output, but no input may have more than one.
The requirement is one-way: it restricts arrows leaving an input, never arrows arriving at an output.
Figure (svg): A function machine labelled with a rule, showing an input going in and exactly one output coming out
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.8 An Introduction to Functions §1.8, pp. 48-48
Section
Section 1
Concept
The whole definition rests on one clause: for each input there is exactly one output. Nothing forbids two inputs from sharing an output, and forgetting that asymmetry is the commonest misunderstanding of the definition.
Figure (svg): Two columns contrasting a pairing that is a function with one that is not
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.8 An Introduction to Functions §1.8, pp. 48-48 — the definition of a function
Picture it
One of these columns breaks the definition and the other does not.
Figure (svg): Two columns contrasting a pairing that is a function with one that is not
The left column fails because asking what the output is for input 2 has no single answer. The right column is fine — a rule may send several inputs to the same place, and squaring does exactly that with 3 and negative 3.
Worked example
Two small pairings, tested against the definition one input at a time.
\[ \text{Pairing A: } 1 \rightarrow 4, \; 2 \rightarrow 5, \; 3 \rightarrow 4. \quad \text{Pairing B: } 1 \rightarrow 4, \; 2 \rightarrow 5, \; 2 \rightarrow 7. \]
Check each input of A in turn
Why: Input 1 has one output, input 2 has one output, input 3 has one output.
Notice that 1 and 3 share the output 4
Why: This is allowed. The definition says nothing about outputs being shared.
Check each input of B in turn
Why: Input 1 has one output, but input 2 has two: 5 and 7.
\[ B:\text{ input } 2\text{ has two} \]
Apply the definition to both
Why: A satisfies the requirement everywhere; B fails at a single input, and one failure is enough.
Figure (svg): The solution to Worked example is this pairing a function shown as a ladder of expressions, one row per algebraic move
\[ \text{A: function} \qquad \text{B: not a function} \]
Verify: ask the rule a question in each case
Why: For A, what is the output when the input is 3? Four — a single, definite answer. For B, what is the output when the input is 2? Five or seven, with nothing to choose between them. A rule that cannot answer its own question is exactly what the definition rules out.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.8 An Introduction to Functions §1.8, pp. 48-48
Sorting
Each item lists the pairs of a small relation. Check each input in turn.
Sort into buckets
Sort each pairing by whether it is a function.
Item e is the square-root relation, and it is exactly why square roots need the positive-only convention you will meet in Chapter 9. Without that convention, taking a square root would not be a function.
Worked example
Example 1 from the textbook. The first six triangular numbers, as a table and as a verdict.
\[ \text{Make an input-output table with input } n \text{ the figure number and output } T \text{ the triangular number, then decide whether it is a function.} \]
Count the dots in each figure
Why: One, three, six, ten, fifteen, twenty-one — each figure adds a new row of dots.
\[ 1, 3, 6, 10, 15, 21 \]
Write the inputs across the top row
Why: The figure numbers 1 through 6.
\[ n: 1 2 3 4 5 6 \]
Write the outputs underneath
Why: Each output sits directly below its own input.
\[ T: 1 3 6 10 15 21 \]
Check the definition input by input
Why: Each of the six inputs appears once and has exactly one output beneath it.
Figure (svg): The first six triangular numbers drawn as growing triangles of dots, with an input-output table beneath
\[ n: 1, 2, 3, 4, 5, 6 \qquad T: 1, 3, 6, 10, 15, 21 \]
Verify: check the pattern between consecutive outputs
Why: The differences are 2, 3, 4, 5 and 6 — each figure adds one more dot than the last addition. That regularity confirms the counting was done correctly, and it is the reason the pattern is called triangular in the first place.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.8 An Introduction to Functions §1.8, pp. 48-48
Trap
\[ 1 \rightarrow 4, \quad 2 \rightarrow 5, \quad 3 \rightarrow 4 \]
Rule this out because the output 4 is used twice
Why: The definition contains the word exactly once, and the eye finds a repeat and stops there.
This is a perfectly good function. The repeat is among the outputs, and the definition places no restriction there at all.
\[ 1 \rightarrow 4, \quad 2 \rightarrow 5, \quad 3 \rightarrow 4 \quad \text{is a function} \]
Check inputs, not outputs — count the arrows leaving each input
Why: The requirement is that the rule can always answer. Sharing an output never stops it answering.
\[ 2 \rightarrow 5 \text{ and } 2 \rightarrow 7 \quad \text{is not a function} \]
A concrete case worth keeping: squaring sends 3 and negative 3 both to 9, and squaring is one of the most important functions in the book.
Elimination
Four statements about the definition of a function. Only one is correct.
Eliminate the wrong options
Which statement is true?
Survives elimination: A
Why: The definition constrains only what leaves an input. Two inputs sharing an output is not merely allowed but extremely common — the squaring function sends 3 and negative 3 both to 9, and a constant rule sends every input to the same output while remaining a perfectly good function.
Notation
Every word in this sentence is doing work. Take them one at a time.
Annotate
On: \( \text{for each input there is } \textbf{exactly one} \text{ output} \)
If you can say why the definition does not mention outputs, you understand it better than most people who have memorised it.
Counterexample
Constructing a counterexample is the fastest way to be sure you have the definition right.
Discussion prompt
Invent a pairing from an everyday situation that is not a function, and say precisely which input breaks it. Then change one thing about your situation so that it becomes a function.
Hint: Look for a situation where one thing can genuinely have two answers.
Answer:
A good example: pairing each student in a class with a sport they play. A student who plays both football and tennis is a single input with two outputs, so the pairing is not a function — asking what sport does this student play has no single answer.
Changing the output to the number of sports a student plays fixes it. Every student plays some definite number of sports, so each input now has exactly one output, and several students sharing the number two is perfectly acceptable. Notice that the fix was to change what the output measures, not to change the students.
Section
Section 2
Concept
One way to describe a function is an input-output table: the inputs across the top row and, directly beneath each one, the output it produces. The alignment is what carries the pairing.
A table can be built from a picture, from a rule, or from a description, and the same table can then be plotted as a graph.
| Input n | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Output T | 1 | 3 | 6 | 10 | 15 | 21 |
Figure (svg): The first six triangular numbers drawn as growing triangles of dots, with an input-output table beneath
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.8 An Introduction to Functions §1.8, pp. 48-48 — Example 1, Make an Input-Output Table
Picture it
Six pictures above, six pairs of numbers below.
Figure (svg): The first six triangular numbers drawn as growing triangles of dots, with an input-output table beneath
The picture holds the meaning and the table holds the pairing. Turning one into the other is the move that makes a pattern into something you can compute with.
Worked example
Example 2, part a. The balloon rises at 20 feet per minute from an altitude of 250 feet.
\[ \text{For } h = 250 + 20t \text{ with } t \text{ from } 0 \text{ to } 5, \text{ make an input-output table.} \]
List the inputs you want
Why: The problem restricts t to between 0 and 5 minutes, so use the six whole minutes.
\[ t: 0 1 2 3 4 5 \]
Substitute each input into the rule and write the working
Why: 250 plus 20 times 0, then times 1, and so on. Writing the substitution line for each keeps the arithmetic checkable.
\[ 250 + 0, 250 + 20,... \]
Simplify each one to get the output
Why: 250, 270, 290, 310, 330 and 350 feet.
\[ h: 250 270 290 310 330 350 \]
Write the outputs directly beneath their inputs
Why: The vertical alignment is what records which output belongs to which input.
Figure (svg): An input-output table for the balloon function h equals 250 plus 20 t, for t from 0 to 5
\[ h = 250 + 20t: \quad 250, \; 270, \; 290, \; 310, \; 330, \; 350 \]
Verify: check the step between consecutive outputs
Why: Every step is exactly 20 feet, which is the rate the problem gave. A constant rate has to produce a constant step, so any irregular gap in the output row would mean an arithmetic slip in that column.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.8 An Introduction to Functions §1.8, pp. 49-49
Faded example
Four of the six outputs are given. Supply the other two.
Fill in the blanks
h = 250 + 20t: \quad t = 0 \rightarrow 250, \; t = 1 \rightarrow 270, \; t = 2 \rightarrow 290, \; t = 3 \rightarrow 310, \; t = 4 \rightarrow 330, \; t = 5 \rightarrow 350
Why: At t equal to zero the rule gives 250 plus nothing, which is the starting altitude. At t equal to five it gives 250 plus 100, which is 350. Both blanks can also be found by continuing the constant step of 20 in either direction, and getting the same answer two ways is a genuine check.
Worked example
No rule is given this time, only words. The table has to come from the meaning.
\[ \text{A taxi charges } 3 \text{ dollars to get in plus } 2 \text{ dollars a mile. Tabulate the fare for } 0 \text{ to } 4 \text{ miles.} \]
Decide what the input and the output are
Why: The input is the number of miles and the output is the fare in dollars.
Work out the output at zero
Why: Zero miles still costs the 3 dollar boarding charge, so the table does not start at zero.
\[ 0\text{ miles gives } 3 \]
Add the per-mile charge for each further mile
Why: Each extra mile adds two dollars: 5, 7, 9, 11.
\[ 3, 5, 7, 9, 11 \]
Write the rule the table implies
Why: Three plus two times the miles, which matches Lesson 1.1's fixed-plus-per-unit shape.
\[ \text{fare } = 2 m + 3 \]
Figure (svg): The solution to Worked example build a table from a description shown as a ladder of expressions, one row per algebraic move
\[ \text{fare} = 2m + 3: \quad 3, \; 5, \; 7, \; 9, \; 11 \]
Verify: check the table against both parts of the description
Why: At zero miles the table gives 3, which is the boarding charge alone, and each step along the table adds 2, which is the per-mile rate. Both numbers from the description appear in the table, one as a starting value and one as a step.
Error analysis
The student tabulated h equals 250 plus 20t for t from 0 to 5. Two entries are wrong.
Annotate
On: \( \begin{array}{c|cccccc} t & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline h & 0 & 270 & 290 & 310 & 330 & 5400 \end{array} \)
The zero case is worth dwelling on. An input of zero almost always deserves separate attention, because it is where the fixed part of a rule appears on its own.
Pattern
Each frame adds one column to the balloon table.
Step through it
What stays the same from one column to the next, and what does that constancy tell you about the graph?
The step of 20 feet per minute never varies, and a constant step is exactly what produces a straight line when the table is plotted. Chapter 4 will call that step the slope.
Translation
Four rules, four output rows for the inputs 0, 1, 2.
Match the pairs
Why: The first three all have constant steps — 2, 2 and 1 respectively — and the value at zero tells you the fixed part of each rule. The fourth has steps of 1 and then 3, which is not constant, and that is exactly what distinguishes a squaring rule from the linear ones. Reading the step and the starting value off a table is how Chapter 4 identifies a rule from its data.
Prediction
The pattern is constant, so the next entry can be predicted before it is computed.
Predict first
The balloon table gives 250, 270, 290, 310, 330, 350 for t from 0 to 5. What would the output be at t equal to 8?
Correct: 410.
\[ h = 250 + 20(8) = 250 + 160 = 410 \text{ feet} \]
Why: Each minute adds 20 feet, so three more minutes past t equal to 5 adds 60 to the 350, giving 410. Substituting directly into the rule confirms it: 250 plus 20 times 8 is 250 plus 160, which is 410. Note that the problem originally restricted t to at most 5, so this is an extrapolation beyond what was modelled — the balloon may well not still be climbing.
Section
Section 3
Concept
The collection of all input values is the domain of the function, and the collection of all output values is the range. Both are read straight off an input-output table.
domain — The collection of all input values of a function. The collection of all output values is called the range.
\[ \text{domain } = \{1, 2, 3, 4, 5, 6\} \qquad \text{range } = \{1, 3, 6, 10, 15, 21\} \]
The domain is the top row of the table and the range is the bottom row, with any repeats listed only once.
Figure (svg): A mapping diagram showing the domain 1 to 6 on the left mapped to the range 1, 3, 6, 10, 15, 21 on the right
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.8 An Introduction to Functions §1.8, pp. 49-49 — the Domain and Range paragraph
Picture it
The left column is the domain, the right column is the range, and the arrows are the function.
Figure (svg): A mapping diagram showing the domain 1 to 6 on the left mapped to the range 1, 3, 6, 10, 15, 21 on the right
Exactly one arrow leaves each item on the left. Nothing is required of how many arrive on the right, which is why the picture is drawn with the arrows pointing one way.
Worked example
Reading both straight off the table from Example 1.
\[ \text{State the domain and range of the function } n \rightarrow T \text{ for the first six triangular numbers.} \]
Read the top row for the domain
Why: The inputs are the figure numbers 1 through 6.
\[ \text{domain } 1\text{ to } 6 \]
Read the bottom row for the range
Why: The outputs are 1, 3, 6, 10, 15 and 21.
\[ \text{range } 1, 3, 6, 10, 15, 21 \]
Check for repeats in the outputs
Why: There are none here, so the range has six members like the domain.
State both as collections
Why: Domain and range are collections of values, not single numbers.
Figure (svg): The solution to Worked example domain and range of the triangular numbers shown as a ladder of expressions, one row per algebraic move
\[ \text{domain } \{1, 2, 3, 4, 5, 6\} \qquad \text{range } \{1, 3, 6, 10, 15, 21\} \]
Verify: count both collections
Why: Six inputs and six outputs, with each input having exactly one output and no two outputs coinciding. Had two inputs shared an output, the range would have been smaller than the domain — which is allowed, and worth noticing when it happens.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.8 An Introduction to Functions §1.8, pp. 49-49
Matching
Four small functions with the domain 1, 2, 3 in every case.
Match the pairs
Why: The third is worth dwelling on: a rule that sends every input to 5 is a perfectly good function, and its range is the single value 5. Three inputs sharing one output is exactly the situation the definition permits, and it is the extreme case of a range smaller than its domain.
Worked example
When outputs repeat, the range has fewer members than the domain.
\[ \text{Find the domain and range of the pairing } \; 1 \rightarrow 4, \; 2 \rightarrow 5, \; 3 \rightarrow 4, \; 4 \rightarrow 5. \]
List the inputs
Why: Four of them: 1, 2, 3 and 4.
\[ \text{domain has } 4\text{ members} \]
List the outputs as they appear
Why: Four, five, four, five.
\[ 4, 5, 4, 5 \]
Remove the repeats
Why: The range is a collection of values, so each distinct value is listed once.
\[ \text{range is } 4\text{ and } 5 \]
Compare the two sizes
Why: Four inputs, two outputs — a perfectly ordinary function.
\[ \text{domain } 4,\text{ range } 2 \]
Figure (svg): The solution to Worked example a range smaller than its domain shown as a ladder of expressions, one row per algebraic move
\[ \text{domain } \{1, 2, 3, 4\} \qquad \text{range } \{4, 5\} \]
Verify: confirm it is still a function
Why: Each of the four inputs appears exactly once and has exactly one output, so the definition is satisfied. The range being smaller than the domain is a consequence of outputs being shared, which the definition explicitly permits.
Trap
\[ 1 \rightarrow 4, \; 2 \rightarrow 5, \; 3 \rightarrow 4, \; 4 \rightarrow 5 \]
Copy the bottom row of the table straight out as the range
Why: The bottom row is right there, and copying it feels like reading rather than deciding.
\[ \text{range } = \{4, 5, 4, 5\} \quad \text{(wrong)} \]
A collection of values lists each distinct value once. Writing 4 twice suggests there are two different fours, which there are not.
\[ \text{range } = \{4, 5\} \]
Read the bottom row, then remove duplicates before writing the range
Why: The range is the collection of values the function can produce, not a transcript of the output row.
The domain is treated the same way, though duplicates cannot arise there — an input appearing twice with the same output is redundant, and appearing twice with different outputs would mean it is not a function at all.
Discrimination
For each description, decide which collection it names.
Sort into buckets
Sort each item by whether it describes the domain or the range.
Edge cases
The balloon problem restricted t to between 0 and 5. Real problems usually restrict their domains.
Discussion prompt
Explain why the balloon function's domain stops at 5, and say what would go wrong with the model if you substituted t equal to 60. Then describe one other everyday function whose domain is restricted by its situation.
Hint: Read the problem again: how long was the burner on?
Answer:
\[ h = 250 + 20t \quad \text{with } t \geq 0 \text{ and } t \leq 5 \]
The burner was on for five minutes, so the rule describes the balloon only during those five minutes. At t equal to 60 the rule would give 1450 feet, which is arithmetic rather than physics — the balloon stopped climbing at minute five, so the model no longer describes anything.
A taxi fare is a good second example: its domain is miles travelled, which cannot be negative and is bounded above by how far the taxi can actually go. Domains restricted by the situation rather than by the arithmetic are the normal case in applied work, and stating them is part of stating the model.
Socratic
The domain and the range need not be the same size, and the reason is built into the definition.
Discussion prompt
Explain why a function's range can have fewer members than its domain but never more. Give one example of each situation you claim is possible.
Hint: Count arrows: how many leave, and how many can arrive at one place?
Answer:
Each input sends out exactly one arrow, so the number of arrows equals the number of inputs. Every arrow lands on some output, so there can never be more distinct outputs than arrows, and therefore never more outputs than inputs.
The range is smaller whenever two arrows land in the same place. The rule sending every input to 5 has a domain of any size and a range of exactly one member. The range equals the domain in size whenever no two arrows share a landing point, as with the triangular numbers, where all six outputs are different.
Section
Section 4
Concept
Once a rule has produced a table, the table can be drawn. The input goes on the horizontal axis, the output on the vertical, each pair becomes a point, and the points are joined if the quantity varies continuously.
Figure (svg): A graph of the balloon's altitude against time, six plotted points joined into a rising straight line
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.8 An Introduction to Functions §1.8, pp. 49-49 — Example 2, part b
Picture it
Six points from the table, joined into a straight line.
Figure (svg): A graph of the balloon's altitude against time, six plotted points joined into a rising straight line
The line is straight because every minute adds the same 20 feet. The graph starts at 250 rather than at the origin because the balloon was already at 250 feet when the burner was lit.
Worked example
Example 2, part b. The table is already built; this is the plotting.
\[ \text{Draw a graph of } h = 250 + 20t \text{ for } t \text{ from } 0 \text{ to } 5. \]
Set up the horizontal axis for the input
Why: Time in minutes, labelled from 0 to 5, which is the domain the problem allows.
\[ t\text{ from } 0\text{ to } 5 \]
Set up the vertical axis for the output
Why: Altitude in feet, labelled from 0 to 400 so that every output from 250 to 350 fits with room to spare.
\[ h\text{ from } 0\text{ to } 400 \]
Plot each pair from the table
Why: To plot the first point, find t equal to 0 on the horizontal axis and h equal to 250 on the vertical, and mark where they meet.
Join the points
Why: Altitude varies continuously as the balloon rises, so the balloon really did have a height at every instant between the readings.
Read the trend off the picture
Why: As time increases the height increases, at a steady rate.
Figure (svg): A graph of the balloon's altitude against time, six plotted points joined into a rising straight line
\[ \text{from } (0, 250) \text{ to } (5, 350) \]
Verify: check the first and last points against the rule
Why: At t equal to 0 the rule gives 250, and the leftmost point sits at 250. At t equal to 5 it gives 350, and the rightmost point sits at 350. Checking the two ends catches almost every plotting error, because a mistake in the scale shows up at the extremes first.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.8 An Introduction to Functions §1.8, pp. 49-49
Elimination
The function is h equals 250 plus 20t.
Eliminate the wrong options
Which of these points lies on the graph?
Survives elimination: A
Why: Substituting t equal to 3 gives 250 plus 20 times 3, which is 310, so the point is (3, 310). Option D is worth naming separately: getting the right two numbers in the wrong order is one of the most common plotting errors, and it is why the horizontal coordinate is always written first.
Worked example
The same function, read for meaning rather than for values.
\[ \text{Use the graph to estimate the altitude at } t = 2.5 \text{ minutes, and say why the estimate is defensible.} \]
Locate 2.5 on the horizontal axis
Why: Halfway between the marks for 2 and 3.
\[ t = 2.5 \]
Read up to the line and across to the vertical axis
Why: The line at that point sits halfway between 290 and 310.
\[ h = 300 \]
Check against the rule
Why: 250 plus 20 times 2.5 is 250 plus 50, which is 300.
\[ h = 300\text{ exactly} \]
Say why reading between the points is legitimate here
Why: Altitude is continuous — the balloon passed through every height on its way up — so the line between two plotted points describes real values.
Figure (svg): The solution to Worked example what the graph tells you that the table does not shown as a ladder of expressions, one row per algebraic move
\[ h = 250 + 20(2.5) = 300 \text{ feet} \]
Verify: compare the graph reading with the rule
Why: Both give exactly 300 feet, which confirms the line was drawn accurately. That agreement is what justifies using a graph to read values the table never listed, and it is why Lesson 1.7's warning about joining unordered categories matters — here the joining is honest.
Trap
\[ h = 250 + 20t \text{ at } t = 0 \]
Draw the line from the origin because the input starts at zero
Why: A graph that starts at the corner looks natural, and zero minutes feels like it should mean zero height.
The balloon was already at 250 feet before the burner was lit. A line from the origin describes a completely different flight, one that begins on the ground.
\[ h = 250 + 20(0) = 250 \;\rightarrow\; \text{the point } (0, 250) \]
Substitute the smallest input into the rule and plot whatever comes out
Why: An input of zero does not imply an output of zero; the fixed part of the rule survives.
The value at zero input is exactly the fixed part of the rule, which is why substituting zero is such a useful check — it isolates the constant from everything else.
Estimation
The line runs from (0, 250) to (5, 350).
Predict first
About what altitude does the graph show at t equal to 4.5 minutes?
Correct: About 340 feet.
\[ h = 250 + 20(4.5) = 250 + 90 = 340 \text{ feet} \]
Why: Four and a half minutes is halfway between 4 and 5, and the outputs there are 330 and 350, so the value is 340. Substituting confirms it: 250 plus 20 times 4.5 is 250 plus 90. Reading between plotted points is legitimate here because altitude changes continuously.
Fill the middle
The first point is being plotted. Supply the two coordinates.
Fill in the blanks
\text250 t = 0: \; h = 250 + 20(0) = 0 \;\rightarrow\; \text250 (___, ___)
Why: Substituting zero leaves the fixed 250 untouched, so the first point is (0, 250). The input is written first in a coordinate pair, which is the convention Chapter 4 will formalise — and getting it backwards is the single most common plotting error, so it is worth building the habit now.
Socratic
Not every function graphs as a straight line. This one does, for a specific reason.
Discussion prompt
Explain what property of the balloon table makes its graph a straight line, and describe what the graph of a table whose steps were 20, 30, 40, 50 would look like instead.
Hint: Look at the differences between consecutive outputs.
Answer:
Every step in the balloon table is exactly 20 feet, so moving one unit right always moves the same distance up. A constant step is precisely what a straight line is, and the size of the step is what Chapter 4 will call the slope.
If the steps grew — 20, then 30, then 40, then 50 — each move to the right would climb further than the last, so the graph would bend upwards into a curve. That is what happens with the triangular numbers, whose steps are 2, 3, 4, 5 and 6, and it is why their graph is not a line.
Section
Section 5
Concept
The same function can be described in four ways, and each one is best at answering a different question. Fluency means being able to start from any of the four and produce the other three.
A question that is hard in one representation is often easy in another, which is why moving between them is worth practising.
| Representation | Best at |
|---|---|
| In words | saying what the function means |
| As a rule | computing any output quickly |
| As a table | giving exact values at chosen inputs |
| As a graph | showing the trend at a glance |
Figure (svg): Four representations of the same function: words, a rule, a table and a graph
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.8 An Introduction to Functions §1.8, pp. 48-49 — the lesson goal, to use four different ways to represent functions
Picture it
All four rows describe the same balloon.
Figure (svg): Four representations of the same function: words, a rule, a table and a graph
Notice that the rule and the graph both contain the 250 and the 20, but in different guises: in the rule they are written down, and in the graph one is where the line starts and the other is how steeply it climbs.
Worked example
A description in English, converted step by step.
\[ \text{A pool contains } 40 \text{ litres and is filled at } 15 \text{ litres per minute. Give the rule, a table and the graph's shape.} \]
Identify the fixed part and the per-unit part
Why: Forty litres is there at the start; fifteen litres arrives each minute.
\[ \text{fixed } 40,\text{ rate } 15 \]
Write the rule
Why: Volume equals forty plus fifteen times the number of minutes.
\[ V = 40 + 15 m \]
Build a short table
Why: At 0, 1, 2 and 3 minutes: 40, 55, 70 and 85 litres.
\[ 40, 55, 70, 85 \]
Describe the graph
Why: A straight line starting at 40 on the vertical axis and rising by 15 for each step right.
\[ \text{line from } (0, 40) \]
Figure (svg): The solution to Worked example start from words and produce the other three shown as a ladder of expressions, one row per algebraic move
\[ V = 40 + 15m \quad \text{at } m = 0, 1, 2, 3: \; 40, \; 55, \; 70, \; 85 \]
Verify: check that all four forms agree at one input
Why: At m equal to 2 the words say forty plus two lots of fifteen, the rule gives 40 plus 30, the table gives 70, and the graph would sit two steps above 40. All four agree, which is what it means for them to be the same function.
Translation
Four descriptions of the same kind of function in different representations.
Match the pairs
Why: All four describe the same balloon, and each contains the same two numbers in different disguises: the 250 is the starting value, the value at input zero, the first table entry and the height where the line meets the vertical axis; the 20 is the rate, the coefficient, the table's step and the line's steepness. Recognising one quantity across four representations is what fluency means here.
Worked example
The reverse direction, which is the one Chapter 4 and Chapter 5 are built on.
\[ \text{A table gives outputs } 7, \; 10, \; 13, \; 16 \text{ for inputs } 0, \; 1, \; 2, \; 3. \text{ Find the rule.} \]
Find the step between consecutive outputs
Why: Ten minus seven is three, thirteen minus ten is three, sixteen minus thirteen is three. The step is constant.
\[ \text{step } = 3 \]
Read the output at the input zero
Why: Seven, which is the fixed part of the rule.
\[ \text{fixed } = 7 \]
Assemble the rule
Why: Fixed part plus step times input.
\[ y = 3 x + 7 \]
Test the rule at an input you did not use to build it
Why: At x equal to 2 the rule gives 6 plus 7, which is 13, matching the table.
Figure (svg): The solution to Worked example start from a table and recover the rule shown as a ladder of expressions, one row per algebraic move
\[ y = 3x + 7 \]
Verify: test the rule at every input in the table
Why: At 0, 1, 2 and 3 the rule gives 7, 10, 13 and 16, which is the whole output row. Recovering a rule from a table is only justified when it reproduces every entry, not just the ones used to find it.
Trap
A table gives 7, 10, 13, 16 for inputs 0, 1, 2, 3.
Notice the step of 3 and the starting value of 7, then write y equals 7x plus 3
Why: Both numbers are correct and their roles get swapped, which is easy when neither is labelled.
\[ \text{At } x = 1: \; 7(1) + 3 = 10 \;\checkmark \qquad \text{At } x = 2: \; 7(2) + 3 = 17 \neq 13 \]
It passes at one input by coincidence and fails everywhere else, which is exactly why one test case is never enough.
\[ y = 3x + 7 \]
The step multiplies the input; the value at input zero stands alone
Why: The step is how much the output changes per unit of input, so it has to be attached to the input.
\[ \text{At } x = 0: \; 7 \quad \text{At } x = 2: \; 13 \quad \text{At } x = 3: \; 16 \;\checkmark \]
Always test a recovered rule at two or more inputs, and include one that is neither the first nor the second.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Question | Best representation | Why |
|---|---|---|
| What is the output at t equal to 37? | the rule | substitute once, no table needed |
| Is the quantity rising or falling? | the graph | the direction is visible at a glance |
| What exactly is the output at t equal to 3? | the table | an exact value, not an estimate |
No single representation wins at everything, which is the reason to keep all four. A question that feels hard is often a question asked of the wrong form.
Reverse engineer
A table's outputs are given for inputs 0, 1, 2 and 3. Recover the rule.
Fill in the blanks
\text4 5, \; 9, \; 13, \; 17 \;\rightarrow\; y = 5x + ___
Why: The differences between consecutive outputs are 4, 4 and 4, so the step is 4 and it multiplies the input. The output at input zero is 5, which stands alone. Testing at input 3 confirms it: four times three plus five is seventeen, which is the last entry in the table.
Socratic
The rule computes everything the others can. It is still not enough on its own.
Discussion prompt
The rule can produce any value in the table and any point on the graph. Give two things the graph shows that the rule does not make obvious, and one thing the words carry that none of the other three do.
Hint: Think about what you notice instantly versus what you have to work out.
Answer:
The graph shows the direction and the steepness immediately — you can see that the balloon is rising and roughly how fast without computing anything. It also shows the whole domain at once, so a restriction such as stopping at five minutes is visible as the line simply ending.
The words carry the meaning: that h is an altitude in feet, that t is minutes since the burner was lit, and that the balloon was already at 250 feet. None of that is recoverable from the symbols alone, which is why every model in this book is expected to come with a sentence saying what its letters stand for.
Comparison
Fill the blanks from memory before you scroll back. The middle row is the one people get backwards.
Comparison matrix
| Situation | A function? | Why |
|---|---|---|
| One input, one output, every time | Yes | the rule can always answer |
| Two inputs sharing one output | Yes | nothing forbids sharing an output |
| One input with two different outputs | No | the rule cannot say which output is meant |
The definition constrains arrows leaving an input and says nothing about arrows arriving at an output. Every question about whether something is a function reduces to that one asymmetry.
Pattern
Whether you are given a picture, a rule, a table or a description, the same five moves cover it.
Step three is checked on inputs and never on outputs. Two inputs sharing an output is allowed, and forgetting that is the commonest error in this lesson.
OpenStax Elementary Algebra 2e, §4.1 Use the Rectangular Coordinate System §4.1
Check
The definition. Check each input in turn.
Check your understanding
Which of these pairings is NOT a function?
Answer: A
Why: In the first pairing the input 1 has two different outputs, 5 and 6, so the rule cannot say what 1 gives. A single ambiguous input disqualifies the whole pairing, whatever the rest of it does.
Check
Building a table. Substitute before you choose.
Check your understanding
For the rule h equals 250 plus 20t, what is the output when t equals 0?
Answer: A
Why: Substituting zero gives 250 plus 20 times 0, which is 250 plus 0, or 250. An input of zero leaves the fixed part of the rule untouched, which is exactly why substituting zero is a good way to isolate that fixed part.
Check
Domain and range. Read them off the table.
Check your understanding
A function has the pairs 1 to 4, 2 to 5, 3 to 4 and 4 to 5. What is its range?
Answer: A
Why: The range is the collection of output values, and the only distinct outputs are 4 and 5. A collection lists each value once, so the repeats are not written again — and here the range has two members while the domain has four, which is exactly what happens when outputs are shared.
Real world
A vending machine has buttons labelled A1 to A6, and each button dispenses a snack. You press A3 twice and get a different snack each time.
Discussion prompt
Say whether the button-to-snack pairing is a function and why, treating the button as the input. Then describe two different real changes to the machine — one that would break the function property and one that would not — and say which quantity you would have to measure instead to restore it.
Hint: Think about what has to be true for the machine to be predictable.
Answer:
A working machine is a function: each button has exactly one snack. Getting a different snack from A3 twice means one input produced two outputs, so this particular machine has stopped being one — and that is precisely what makes it feel broken, since you can no longer predict what pressing a button will do.
Two buttons dispensing the same snack would not break anything: two inputs may share an output, and a machine stocking the same crisps in A1 and A4 is perfectly predictable. What breaks it is one button with two possible outcomes.
To restore a function you could take the input to be the pair of button and time, or measure a quantity that is genuinely determined by the button alone, such as its price. The general move is to enrich the input until it determines the output, and it is the same move that makes square roots a function by taking only the positive value.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
If two different inputs give the same output, is the pairing still a function?
Correct: Yes — nothing forbids two inputs sharing an output.
\[ 3 \rightarrow 9 \quad \text{and} \quad -3 \rightarrow 9 \quad \text{— still a function} \]
\[ 2 \rightarrow 5 \quad \text{and} \quad 2 \rightarrow 7 \quad \text{— not a function} \]
Why: The definition constrains what leaves an input, not what arrives at an output. Squaring sends both 3 and negative 3 to 9 and is one of the most important functions in the course; a rule sending every input to the same output is also a function, with a range of exactly one value. Reading the requirement backwards is the single most common misunderstanding of this definition, and it is worth checking yourself on deliberately.
Explain it
They are comfortable with formulas and have never heard the word function.
Discussion prompt
In no more than four sentences, explain what a function is without using the words domain or range. Then give them one test they can apply to any table to decide whether it shows a function, and name the mistake the test is designed to prevent.
Hint: Your test should be about looking down a particular row.
Answer:
A usable answer: a function is a rule that always gives one definite answer. You feed it a number and it hands one back, and it hands back the same one every time you feed it that number. It is allowed to give the same answer for two different inputs — what it may never do is give two answers for the same input.
The test: look along the input row for any value that appears twice. If one does, check whether its two outputs agree; if they differ, it is not a function. The mistake this prevents is checking the output row instead, which would wrongly reject every rule that sends two inputs to the same place — including squaring.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: The function test is fixed by always checking inputs and never outputs, and by keeping the squaring example in mind as proof that shared outputs are fine. Tables are fixed by writing the substitution line for every column and checking the step between them. Domain and range are fixed by reading the top row and the bottom row and removing duplicates from the second. Plotting is fixed by writing the input coordinate first, every time, and by substituting zero to find where the graph starts. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
Divide a page into four quadrants and put one function in all four of them: write it in words in the first, as a rule in the second, as an input-output table in the third, and as a graph with both axes labelled in the fourth. Choose a function with a fixed part and a steady rate, so that the same two numbers appear in every quadrant, and circle those two numbers wherever they occur. Underneath, write the domain and the range of your function as two collections. Finally, in the margin, draw one small pairing that is not a function and mark with an arrow the exact input that breaks it.
The two circled numbers should appear four times each, once per quadrant. If one of them is missing from the graph, look again at where the line meets the vertical axis and at how steeply it climbs.
Recap
Five things, and the first one is the definition the rest of the book is built on.
| If the question says | Your first move is |
|---|---|
| Does the table represent a function | Look for a repeated input, not a repeated output |
| Make an input-output table | Substitute each input and write the working |
| State the domain and range | Read the two rows, then remove duplicates |
| Use the table to draw a graph | Input on the horizontal axis, output on the vertical |
| Find the rule from the table | Find the step, then the output at zero |
That completes Chapter 1. Chapter 2 goes back to the number line and builds the arithmetic of negative numbers properly, so that the expressions you have been evaluating can have any real number substituted into them rather than only the convenient ones.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.8 An Introduction to Functions §1.8, pp. 48-53 — everything on these slides traces back here
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