Relations as any set of ordered pairs, and functions as the relations in which every input has exactly one output. Includes the vertical line test, function notation f of x with its evaluation by substitution, and linear functions of the form f of x equals mx plus b graphed from the slope-intercept form.
Subject: Algebra 1 · 65 slides · symbolic lesson
Open the interactive version of this deck
Title
Algebra 1 · Chapter 4 — Graphing Linear Equations and Functions
Functions and Relations
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. 4 Graphing Linear Equations and Functions — Lesson 4.8 Functions and Relations §4.8, pp. 252-258 — the lesson these objectives are drawn from
Warm-up
Lesson 1.8 defined a function and Lesson 4.3 found a graph that failed the definition. This lesson names the wider class those failures belong to.
Discussion prompt
The pairs (0, 0), (1, 1), (4, 2) and (4, -2) all satisfy the equation x equals y squared. Is this collection a function of x, and what exactly goes wrong?
Hint: Look at how many outputs the input four has.
Answer:
\[ x = 4: \; y = 2 \text{ and } y = -2 \]
The input four has two different outputs, so the definition from Lesson 1.8 fails. The collection of pairs is still a perfectly good object — it is called a relation — and being a function is an extra condition that this one does not meet.
Concept
A relation is any set of ordered pairs. A relation is a function if for every input there is exactly one output. So functions are a subset of relations, singled out by that one requirement.
relation — Any set of ordered pairs. A relation is a function if for every input there is exactly one output.
The failure is always on the input side: one input with two outputs.
Figure (svg): Two input-output diagrams, one a function and one only a relation
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.8 Functions and Relations §4.8, pp. 252-252
Section
Section 1
Concept
Any collection of ordered pairs is a relation. It is a function when no input appears twice with different outputs. Two inputs sharing an output is allowed and always has been.
The equation x equals y squared gives the pairs (0, 0), (1, 1), (4, 2) and (4, -2), which form a relation and not a function.
Figure (svg): The relation x equals y squared, showing one input with two outputs
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.8 Functions and Relations §4.8, pp. 252-252 — the paragraph defining relation and function
Picture it
One input, two outputs.
Figure (svg): The relation x equals y squared, showing one input with two outputs
The dashed vertical line at four meets the curve twice, which is the picture of the input four having two outputs. That observation is about to become a formal test.
Worked example
This is Example 1 from the textbook. Two input-output diagrams.
\[ \text{(a) } 1\to 2, \; 2\to 4, \; 3\to 4, \; 4\to 5. \quad \text{(b) } 1\to 5, \; 1\to 7, \; 4\to 9. \]
Check each input in the first
Why: Every input has exactly one arrow leaving it.
Give its domain and range
Why: The domain is 1, 2, 3 and 4; the range is 2, 4 and 5.
Notice the repeated output
Why: Both 2 and 3 map to 4, which the definition allows.
Check the second
Why: The input 1 has two outputs, 5 and 7.
Figure (svg): Two input-output diagrams, one a function and one only a relation
\[ \text{(a) domain } \{1,2,3,4\}, \text{ range } \{2,4,5\} \]
Verify: check that the range lists each value once
Why: Four appears as an output twice and is listed once in the range, because a range is a set of values rather than a tally. Listing it twice would suggest something was wrong with the function, and nothing is.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.8 Functions and Relations §4.8, pp. 252-252
Sorting
Look for an input with two different outputs.
Sort into buckets
Sort each set of pairs by whether it is a function of x.
Every rejected set fails for the same reason and every accepted one passes for the same reason. There is only one condition to check, and it is always about the first coordinates.
Worked example
A table is a relation written in rows.
\[ \text{Is } \; (0, 0), \; (1, 1), \; (4, 2), \; (4, -2) \; \text{ a function of } x? \]
List the inputs
Why: Zero, one, four and four.
\[ 0, 1, 4, 4 \]
Look for a repeated input
Why: Four appears twice.
\[ 4\text{ repeats} \]
Compare its two outputs
Why: They are two and negative two, which differ.
State the verdict
Why: The relation is not a function of x.
Figure (svg): The solution to Worked example the same test on a table shown as a ladder of expressions, one row per algebraic move
\[ (4, 2) \text{ and } (4, -2) \]
Verify: ask whether swapping the roles would help
Why: Reading the pairs the other way round, as y determining x, every input would have exactly one output — zero gives zero, one gives one, two gives four and negative two gives four. So this relation is a function of y even though it is not a function of x, which shows that the question always has to name which variable is the input.
Trap
The relation sends 2 to 4 and 3 to 4.
Rule it out, since the output 4 is used twice
Why: Reusing a value looks like the kind of duplication the definition forbids.
The definition restricts how many outputs an input may have, and says nothing about how many inputs may share an output. Two arrows arriving is fine; two arrows leaving is not.
The relation is a function. Each of 2 and 3 has exactly one output, and they happen to be the same.
Check inputs, never outputs
Why: The asymmetry is deliberate and it is what makes constant functions from Lesson 4.3 legitimate.
A constant function sends every input to one output, which is the most extreme version of this and is still a function.
Faded example
List each value once.
Fill in the blanks
For (1, 2), (2, 4), (3, 4), (4, 5): the domain is 1, 2, 3, 4 and the range is 2, 4, 5.
Why: The domain lists every input and the range every output, each written once. Four appears in both lists here for different reasons: it is an input of the relation and also the output shared by the inputs two and three.
Elimination
Check the first coordinates for repeats.
Eliminate the wrong options
Which of these is NOT a function of x?
Survives elimination: A
Why: The input three appears twice with different outputs, so no single output can be named for it. Option B is the mirror image and is perfectly fine, which is the distinction worth holding on to: the definition constrains arrows leaving an input and never arrows arriving at an output.
Socratic
The definition could have forbidden repeated outputs too.
Discussion prompt
Explain why the definition of a function restricts inputs but not outputs. Then say what you would lose if repeated outputs were forbidden as well.
Hint: Ask what a function is supposed to do.
Answer:
A function is a rule for producing an output from an input, so it has to give one answer to the question what is the output at this input. An input with two outputs makes that question unanswerable. An output reached from two inputs makes no question unanswerable at all — nothing is being asked in that direction.
Forbidding repeated outputs would exclude every constant function, including the horizontal lines of Lesson 4.3, and would exclude a function like y equals x squared, where two and negative two both give four. Those are useful and unremarkable rules, so a definition excluding them would be the wrong definition. Relations that also have distinct outputs for distinct inputs do have a name — they are called one-to-one — but that is a further condition rather than part of being a function.
Section
Section 2
Concept
A graph represents a function if no vertical line intersects it at more than one point. Each vertical line is one input, and two crossings on it would be two outputs for that input.
The textbook suggests holding a pencil upright and passing it across.
Figure (svg): Four graphs judged by the vertical line test
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.8 Functions and Relations §4.8, pp. 253-253 — the Vertical Line Test for Functions summary and Example 2
Picture it
Two pass and two fail.
Figure (svg): Four graphs judged by the vertical line test
The third is the vertical line from Lesson 4.3, which fails because the vertical line test's own line lies right on top of it. The fourth is a sideways curve, failing in the same way at every input it covers.
Worked example
This is Example 2 from the textbook.
\[ \text{Use the vertical line test on (a) a slanted line and (b) a sideways curve.} \]
Sweep a vertical line across the first
Why: It meets the line at exactly one point wherever it is placed.
Conclude for the first
Why: No vertical line meets it more than once, so it is a function.
Sweep across the second
Why: A vertical line can be placed so that it meets the curve twice.
Conclude for the second
Why: That input has two outputs, so it is not a function.
Figure (svg): Four graphs judged by the vertical line test
\[ \text{(a) passes} \qquad \text{(b) fails} \]
Verify: name the two outputs the failing graph gives
Why: Reading the two crossing points off gives the input's two output values, and quoting them is a stronger answer than saying it fails the test. The test is a shortcut for the definition, and being able to fall back on the definition is what makes the shortcut safe to use.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.8 Functions and Relations §4.8, pp. 253-253
Discrimination
Sweep a vertical line across each in your head.
Sort into buckets
Sort each graph by whether it represents a function of x.
Worked example
The horizontal and vertical lines, judged by the test.
\[ \text{Apply the vertical line test to } \; y = 2 \; \text{ and } \; x = -3. \]
Sweep across y equals 2
Why: Every vertical line crosses the horizontal line exactly once.
Conclude
Why: It is a function — a constant function.
Sweep across x equals negative 3
Why: The vertical line placed at negative three lies along the graph.
Conclude
Why: That input has infinitely many outputs, so it is not a function.
Figure (svg): The solution to Worked example two graphs from Lesson 4.3 shown as a ladder of expressions, one row per algebraic move
\[ y = 2: \;\checkmark \qquad x = -3: \;\times \]
Verify: check this against Lesson 4.3
Why: That lesson reached the same two verdicts by listing outputs rather than by drawing lines, so the two methods agree. Agreement between a picture-based test and a definition-based argument is exactly what you want before trusting the picture.
Error analysis
The student was testing a graph shaped like a U opening upwards.
Annotate
On: \( \begin{aligned} &\text{A horizontal line crosses it twice.} \\ &\text{Therefore it is not a function.} \end{aligned} \)
Which line to sweep follows from which axis carries the input. The input is on the horizontal axis, so one input is a vertical line, and that is the only line the test can use.
Prediction
The input is on the horizontal axis.
Predict first
Why does the test use vertical lines rather than horizontal ones?
Correct: A vertical line collects all the outputs belonging to one input.
\[ x = a \text{ is the set of points whose input is } a \]
Why: One input is one x-value, and the set of points with that x-value is precisely a vertical line. So counting crossings on it counts that input's outputs, which is exactly what the definition asks about. A horizontal line would count how many inputs share an output, which the definition does not restrict at all — so the choice is forced by the definition rather than conventional.
Faded example
Fill in the two decisive words.
Fill in the blanks
A graph is a function if no vertical line meets it at more than one point.
Why: Both blanks come straight from the definition: one input is a vertical line, and exactly one output means at most one crossing. Recovering the test from the definition is more reliable than memorising it, since the direction is easy to misremember.
Socratic
The other sweep is not useless — it just answers a different question.
Discussion prompt
Say what it means when a horizontal line crosses a graph twice, and name a graph where this happens. Then say why this is not a problem for being a function.
Hint: Two crossings on a horizontal line means two points sharing what?
Answer:
Two crossings on a horizontal line means two different inputs produce the same output. On the graph of y equals x squared the line at height four crosses at negative two and at two, since both square to four.
The definition of a function says nothing about outputs being shared, so this is permitted. It does mean the function is not one-to-one, which matters when you want to run it backwards — knowing the output is four does not tell you the input. Chapter 9 will meet this again when solving quadratic equations, where both roots have to be reported.
Section
Section 3
Concept
When a function is defined by an equation it is often convenient to name it. Writing f of x in place of y gives function notation, in which the symbol records both which rule is being used and what is going into it.
\[ y = 3x - 2 \;\Longleftrightarrow\; f(x) = 3x - 2 \]
The symbol is read the value of f at x, or f of x. It does not mean f times x.
Figure (svg): The same rule written in x-y notation and in function notation
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.8 Functions and Relations §4.8, pp. 254-254 — the Function Notation paragraph and the Reading Algebra note
Picture it
The same equation written twice.
Figure (svg): The same rule written in x-y notation and in function notation
Any letter may name a function, just as any letter may be a variable. The textbook uses f, g and h, and the choice carries no meaning of its own.
Worked example
This is Example 3 from the textbook.
\[ \text{Evaluate } \; f(x) = 2x + 3 \; \text{ when } x = 2. \]
Write the original function
Why: The rule is two x plus three.
\[ f(x) = 2 x + 3 \]
Substitute 2 for x
Why: Every x in the rule is replaced.
\[ f(2) = 2(2) + 3 \]
Simplify
Why: Four plus three is seven.
\[ 7 \]
Read the notation back
Why: f of 2 equals 7 says the output at input two is seven.
\[ f(2) = 7 \]
Figure (svg): Evaluating a function at a value by substitution
\[ f(2) = 2(2) + 3 = 7 \]
Verify: compare with the x-y notation
Why: In x-y notation this says the point (2, 7) is on the graph, and substituting two into y equals 2x plus 3 gives seven. The two notations describe the same computation, and f of 2 equals 7 packs the whole ordered pair into one line.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.8 Functions and Relations §4.8, pp. 254-254
Notation
Each part of the symbol carries information.
Annotate
On: \( f(x) = 3x - 2 \)
Once the notation is read correctly, evaluating is exactly the substitution from Lesson 1.4. The novelty is in the writing rather than in the arithmetic.
Worked example
Checkpoints 3 and 4. One rule is linear and one is not.
\[ \text{Evaluate } \; f(x) = 4x - 5 \text{ at } x = 2, \text{ and } \; g(x) = x^2 \text{ at } x = 3. \]
Substitute into f
Why: Four times two is eight, minus five.
\[ f(2) = 3 \]
Substitute into g
Why: Three squared is nine.
\[ g(3) = 9 \]
Compare the two rules
Why: The first has the form mx plus b and the second does not.
Note the different names
Why: Using f and g keeps the two rules apart in the same discussion.
Figure (svg): The solution to Worked example two more from guided practice shown as a ladder of expressions, one row per algebraic move
\[ f(2) = 3 \qquad g(3) = 9 \]
Verify: check that the names are doing real work
Why: Writing both as y equals something would make it impossible to say which rule was meant when two are in play. The names f and g are what let a single sentence refer to two different functions, which is the practical reason the notation exists.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.8 Functions and Relations §4.8, pp. 254-254
Trap
\[ f(x) = 2x + 3 \quad \text{find } f(2) \]
Treat f(2) as f times 2 and try to find the value of f
Why: Two symbols written side by side with brackets usually does mean multiplication.
There is no number called f. The letter names the rule, and the bracket holds the input rather than a factor.
\[ f(2) = 2(2) + 3 = 7 \]
Read the bracket as holding the input and substitute it for x
Why: The symbol is read f of two, not f times two.
The textbook's Reading Algebra note says this explicitly, and it is worth saying the words aloud the first few times.
Faded example
Replace every x with the given number.
Fill in the blanks
f(x) = 2x + 3, \; f(2) = 2(2) + 3 = 7
Why: The number in the brackets goes wherever x appears in the rule, giving four plus three, which is seven. In x-y notation this is the point (2, 7), so one line of function notation carries a whole ordered pair.
Elimination
The function is f(x) = 4x - 5.
Eliminate the wrong options
Which statement is correct?
Survives elimination: A
Why: Substituting three gives twelve minus five, which is seven. Option C is the one worth dwelling on, because it names a genuinely useful question — the reverse one — and distinguishing evaluate from solve is what stops the two being confused later.
Socratic
The equation y equals 3x minus 2 says the same thing.
Discussion prompt
Explain what function notation records that x-y notation does not. Then give a situation in which the difference actually matters.
Hint: Imagine two functions in one discussion.
Answer:
Function notation records which rule is being used and which input, in one symbol. In x-y notation both facts have to be carried in the surrounding sentence: y equals eleven leaves open which equation produced it and at what value of x.
It matters as soon as more than one function is in play. Comparing two cost models, you can write f of 35 and g of 35 side by side and the meaning is unambiguous; with two equations both saying y equals something, every sentence has to explain which y it means. It also makes a whole ordered pair fit on one line, which is why the notation takes over from here on.
Section
Section 4
Concept
A function is called linear if it has the form f of x equals mx plus b. To graph one, rewrite it in x-y notation and use the slope and y-intercept from Lesson 4.7.
\[ f(x) = \tfrac{1}{2}x - 3 \;\Longleftrightarrow\; y = \tfrac{1}{2}x - 3 \]
A function like g of x equals x squared is not linear, since it is not of that form.
Figure (svg): The graph of a linear function written in function notation
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.8 Functions and Relations §4.8, pp. 254-254 — the definition of a linear function and Example 4
Picture it
Rewrite, then read off the two numbers.
Figure (svg): The graph of a linear function written in function notation
The graph is the same one Lesson 4.7 would have drawn. Changing the notation on the left of the equation changes nothing about the picture.
Worked example
This is Example 4 from the textbook.
\[ \text{Graph } \; f(x) = \tfrac{1}{2}x - 3. \]
Rewrite in x-y notation
Why: Replace f of x with y.
\[ y = (\frac{1}{2}) x - 3 \]
Find the slope and intercept
Why: One half and negative three.
\[ m = \frac{1}{2}, b = -3 \]
Plot the intercept and step
Why: From (0, -3) move up one and right two.
\[ (2, -2) \]
Draw the line
Why: One straight line through both points.
Figure (svg): The graph of a linear function written in function notation
\[ \text{through } (0, -3) \text{ and } (2, -2) \]
Verify: check the second point using the function notation
Why: Evaluating f of 2 gives one minus three, which is negative two, matching the point the step reached. Doing the check in function notation rather than in x-y notation is good practice, since the notation is about to become the standard one.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.8 Functions and Relations §4.8, pp. 254-254
Sorting
Check whether the rule has the form mx plus b.
Sort into buckets
Sort each function by whether it is linear.
The constant function is the boundary case, and it is worth deciding deliberately rather than by instinct: m equals zero is allowed by the form, so h of x equals 7 is linear.
Worked example
The test is the form rather than the letter used to name it.
\[ \text{Which are linear? } \; f(x) = 4x - 5, \quad g(x) = x^2, \quad h(x) = -3x + 1. \]
Check the first
Why: It has the form mx plus b with m four and b negative five.
Check the second
Why: The input is squared, which the form does not allow.
Check the third
Why: It has the form with m negative three and b one.
Note what the name does not tell you
Why: The letters f, g and h carry no information about the form.
Figure (svg): The solution to Worked example which functions are linear shown as a ladder of expressions, one row per algebraic move
\[ f(x) = 4x - 5 \text{ and } h(x) = -3x + 1 \text{ are linear} \]
Verify: check the second by tabulating it
Why: Squaring the inputs one, two and three gives one, four and nine, and the steps between them are three and then five rather than constant. A non-constant step means the graph bends, so g cannot be linear — exactly the argument from Lesson 4.2.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.8 Functions and Relations §4.8, pp. 254-254
Trap
\[ h(x) = -3x + 1 \]
Treat this as a different kind of object because it is called h rather than f
Why: The letter f has appeared in every example so far, so a different letter looks like a different thing.
The name is arbitrary. This is a linear function with slope negative three and y-intercept one, and it is graphed exactly like any other.
\[ h(x) = -3x + 1 \;\Longleftrightarrow\; y = -3x + 1 \]
Read past the name to the form
Why: Whether a function is linear depends on the shape of the rule, not on what it is called.
The textbook's Study Tip says this directly: just as any letter can be a variable, any letter can name a function.
Faded example
Function notation to x-y notation.
Fill in the blanks
f(x) = \tfrac2-3x - 3 \;\Longrightarrow\; y = \tfrac______x - 3, \quad m = \tfrac______}, \; b = ___
Why: Replacing f of x with y changes nothing but the label on the left, so the slope and intercept are read off exactly as in Lesson 4.7. The step is up one and right two, landing on (2, -2).
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| y = 3x - 2 | f(x) = 3x - 2 | |
|---|---|---|
| Names the rule | no | yes |
| Shows the input | only in the surrounding sentence | in the brackets |
| The graph | a line of slope 3 | the same line |
The last row is the point of the comparison: nothing about the mathematics changes. The notation is a bookkeeping improvement, and it matters most when several functions are in play at once.
Socratic
And is every line the graph of a linear function?
Discussion prompt
Say whether both directions of that claim hold, and give the exception if there is one.
Hint: Think about the lines from Lesson 4.3.
Answer:
Every linear function does graph as a straight line: f of x equals mx plus b is the slope-intercept form, so its graph is the line with slope m and intercept b, including the horizontal case where m is zero.
The reverse fails for vertical lines. The line x equals negative three is straight and is not the graph of any function of x, since it fails the vertical line test. So linear functions correspond to all the lines except the vertical ones — which is the same exception that has appeared in every lesson of this chapter, and it always comes from the same place.
Section
Section 5
Concept
A non-vertical line can be described as a set of ordered pairs, as an equation in several forms, as a graph, and as a function evaluated at inputs. All four describe the same object.
Moving between these four is what the whole chapter has been building.
Figure (svg): The graph of a linear function written in function notation
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.8 Functions and Relations §4.8, pp. 252-255
Picture it
Pairs, equation, graph, rule.
Figure (svg): The graph of a linear function written in function notation
A question that looks hard in one description is often easy in another. Recognising which one the question is phrased in, and which one answers it, is most of the work.
Worked example
The line through (0, -3) with slope one half.
\[ \text{Describe this line as pairs, as an equation, as a graph and as a function.} \]
As ordered pairs
Why: It contains (0, -3), (2, -2), (4, -1) and endlessly many more.
As an equation
Why: In slope-intercept form, and in standard form as x minus 2y equals 6.
As a graph
Why: A straight line rising one for every two across, crossing the axes at 6 and negative 3.
As a function
Why: f of x equals one half x minus three, with f of 4 equal to negative one.
Figure (svg): The graph of a linear function written in function notation
\[ f(x) = \tfrac{1}{2}x - 3 \;\Longleftrightarrow\; x - 2y = 6 \]
Verify: check one pair against all four descriptions
Why: The pair (4, -1) appears in the list, satisfies both equations since four minus negative two is six, sits on the drawn line, and is what f of 4 evaluates to. All four descriptions agree on it, which is what makes them descriptions of one object rather than four.
Matching
Each question is easiest in one particular description.
Match the pairs
Why: Each form makes one fact free and the others cost a little work. Every question here can be answered in any of the forms, and the point of matching them is that choosing well turns several steps into one.
Worked example
Different questions are easiest in different descriptions.
\[ \text{Which description answers each question most quickly?} \]
Where does it cross the axes
Why: Standard form, by setting each variable to zero in turn.
Is it parallel to another line
Why: Slope-intercept form, by comparing slopes.
What is the output at 40
Why: Function notation, by evaluating f of 40.
Is (4, -1) on it
Why: Any equation, by substituting the pair.
Figure (svg): The solution to Worked example choose the right description shown as a ladder of expressions, one row per algebraic move
\[ \text{intercepts} \to \text{standard}, \quad \text{parallel} \to \text{slope-intercept} \]
Verify: try one of them in the wrong description
Why: Finding the intercepts from function notation means solving f of x equals 0 and evaluating f of 0 — two computations rather than the two one-step equations standard form gives. The answer is the same either way, and the effort is not, which is the whole reason for keeping several forms available.
Trap
A student learns tables, intercepts, slope, direct variation and function notation as five separate procedures to memorise.
Practise each one in its own exercise set and never convert between them
Why: Each section of the chapter had its own heading and its own worked examples.
A question phrased in one description then has to be answered in that description, even when another would settle it in one line.
Convert first, then answer
Why: Rewriting into whichever form makes the question easy is usually faster than answering it in the form it arrived in.
A question about parallelism given two standard-form equations is three lines of rewriting and one comparison, rather than any graphing at all.
Elimination
Three of these describe the line through (0, -3) with slope one half.
Eliminate the wrong options
Which one is a different line?
Survives elimination: A
Why: This one has slope two rather than one half, so it passes through the right intercept and tilts far more steeply. The intercept matching is what makes it a plausible distractor, and it is a reminder that a line needs both numbers to be identified.
Hypothesis
Predict before you check.
Predict first
Which of these cannot be written in the form f of x equals mx plus b?
Correct: x = -3.
\[ y = -3: \; f(x) = -3 \qquad y = 2x: \; f(x) = 2x + 0 \]
Why: A vertical line has no y in its equation to isolate, so no rule can produce an output from an input — and it fails the vertical line test at the single input negative three. The second is the constant function f of x equals negative three, which is the case m equals zero; the third is a direct variation model with b equal to zero; and the fourth rearranges to y equals negative two thirds x plus two. Only the vertical line is excluded, and it is the same exception that has recurred throughout the chapter.
Socratic
Eight lessons, and one idea underneath them.
Discussion prompt
In your own words, say what connects the coordinate plane, tables, intercepts, slope, direct variation, slope-intercept form and functions. Then name the one object that keeps failing to fit.
Hint: Ask what all eight lessons were describing.
Answer:
They are all descriptions of the same thing: a linear relationship between two quantities. Lesson 4.1 gave a place to draw it, 4.2 turned an equation into a picture, 4.3 to 4.4 found the landmarks, 4.5 measured its steepness, 4.6 handled the case through the origin, 4.7 packed the whole line into two numbers, and 4.8 named the rule.
The object that keeps failing to fit is the vertical line. It has no slope, no y-intercept, no function form, and it fails the vertical line test — and every one of those failures comes from the same source, that its run is zero. Noticing that a single fact explains four separate exceptions is worth more than remembering the four exceptions.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Relation | Function | |
|---|---|---|
| What it is | any set of ordered pairs | a relation with one output per input |
| Repeated inputs | allowed | allowed only with the same output |
| Repeated outputs | allowed | allowed |
| Graph test | none needed | the vertical line test |
Only one row differs between the columns, and it is the row about inputs. Everything else about the two ideas is identical.
Pattern
Whether the question gives you pairs, a graph or a rule, the same five moves cover it.
Step one is easy to skip and it matters: x equals y squared is not a function of x and is a function of y, so the same set of pairs gives opposite answers depending on which variable is the input.
OpenStax Intermediate Algebra 2e, §3.5 Relations and Functions §3.5
Check
Look for a repeated input.
Check your understanding
Which set of pairs is NOT a function of x?
Answer: A
Why: The input two has two different outputs, three and seven, so no single output can be named for it and the definition fails.
Check
Substitute the bracketed value.
Check your understanding
If f(x) = 4x - 5, what is f(2)?
Answer: A
Why: Substituting two gives eight minus five, which is three. In x-y notation this says the point (2, 3) is on the graph of the function.
Check
Sweep the right line.
Check your understanding
Which graph fails the vertical line test?
Answer: A
Why: A vertical line through the middle of a circle meets it at two points, so that input has two outputs and the circle is not a function of x.
Real world
A monarch butterfly migrating at a steady speed covers a distance d that depends on the travelling time t.
Discussion prompt
Write this as a function using function notation, say why distance is a function of time rather than the other way round being automatic, and say what the domain should be.
Hint: Ask whether one moment can correspond to two distances.
Answer:
\[ d(t) = rt \quad \text{where } r \text{ is the steady speed} \]
At any one moment the butterfly has travelled exactly one distance, so each input time gives exactly one output distance and the relation is a function of time. It is also a direct variation model from Lesson 4.6, since no distance has been covered at time zero.
The domain should be the times from zero to the end of the migration. Negative times describe moments before the journey started and times beyond the end are outside what the model was built for — a restriction worth stating, exactly as with the balloon in Lesson 1.8 and the phone card in Lesson 4.4.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Is the relation (1, 4), (2, 4), (3, 4) a function?
Correct: Yes, because each input has exactly one output.
\[ 1 \to 4, \quad 2 \to 4, \quad 3 \to 4 \quad \text{one arrow leaves each input} \]
\[ \text{compare } 1 \to 4 \text{ and } 1 \to 7 \quad \text{two arrows leave } 1 \]
Why: The definition restricts how many outputs an input may have and says nothing about how many inputs may share an output. Each of one, two and three has exactly one output here, so the relation is a function — a constant function, which is the graph of the horizontal line y equals 4 from Lesson 4.3. The instinct that repetition must be a problem is the single most common error with this definition, and it is worth checking which side of the arrow the repetition is on before deciding.
Explain it
They have met functions once and think any repetition disqualifies them.
Discussion prompt
In no more than four sentences, explain the difference between a relation and a function without repeating the textbook's wording. Then give them the picture that makes the vertical line test obvious.
Hint: Think about arrows leaving and arriving.
Answer:
A usable answer: a relation is just a list of paired-up numbers, with no rules at all. It counts as a function when you can always answer the question given this input, what is the output — which fails only if some input has been given two different answers. Two inputs landing on the same answer is fine, because nothing is ambiguous there.
For the picture, one input is one vertical line on the graph, since that is where all the points with that x-value live. So if a vertical line hits the graph twice, that input has two answers and the graph is not a function. That is the whole test, and it comes straight from the definition rather than being a separate rule.
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 repetition question is fixed by checking arrows leaving an input and never arrows arriving. The test direction is fixed by remembering that one input is one vertical line, so vertical is forced. The notation is fixed by saying f of x aloud and remembering there is no number called f. Graphing is fixed by rewriting into y equals mx plus b first, after which it is Lesson 4.7 unchanged. 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
Draw two input-output diagrams side by side, one a function and one not, with arrows, and beside each write one sentence saying which condition it meets or fails. Underneath, write the domain and range of the one that is a function, listing each value once. In the middle of the page draw four small graphs — a slanted line, a horizontal line, a vertical line and a circle — and beside each draw a dashed vertical line and write whether it passes the test. In the lower half, write one linear function in function notation, evaluate it at two inputs showing the substitutions, rewrite it in x-y notation, and graph it using its slope and intercept. Finally, in the margin, write one sentence explaining why the test uses vertical lines and not horizontal ones.
Your vertical line should meet three of the four small graphs exactly once. If it meets the circle only once, redraw it through the middle rather than at the edge — the test asks whether any vertical line meets it twice, not whether every one does.
Recap
Five things, and the first is the one people get backwards.
| If the question says | Your first move is |
|---|---|
| Is this relation a function | Look for one input with two outputs |
| Give the domain and range | List every input, then every output, once each |
| Does this graph represent a function | Sweep a vertical line and count crossings |
| Evaluate f(3) | Substitute 3 for every x in the rule |
| Graph f(x) = mx + b | Rewrite as y = mx + b, then plot and step |
That completes Chapter 4. Chapter 5 turns the question round: instead of drawing a line from its equation, it finds the equation of a line from information about it — from its slope and a point, from two points, or from a table of data.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.8 Functions and Relations §4.8, pp. 252-258 — everything on these slides traces back here
Want this taught 1-on-1? Alexander tutors Algebra 1 — $55/session, free consultation.