Relations and their four representations, domain and range, the definition of a function and the vertical line test, equations in two variables and their graphs, and function notation for linear and non-linear functions.
Subject: Algebra 2 · 65 slides · symbolic lesson
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Title
Algebra 2 · Chapter 2 — Linear Equations and Functions
Represent Relations and Functions
Objectives
Five outcomes. The second is a definition you will be applying for the rest of the course.
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 72-79 — the lesson these objectives are drawn from
Warm-up
You have been substituting values into expressions since Lesson 1.2. This lesson gives that process a name and a picture.
Discussion prompt
Take the expression 3x minus 5. Substitute x equal to 2, then x equal to 0, then x equal to negative 1. What do those three results have in common as a set, and what would it mean if one input gave you two different results?
Hint: Think about whether substitution could ever be ambiguous.
Answer:
\[ 3(2) - 5 = 1 \qquad 3(0) - 5 = -5 \qquad 3(-1) - 5 = -8 \]
Each input produced exactly one output — arithmetic is not ambiguous. That property has a name: the rule is a function. Relations that fail it exist, but they never come from substituting into a single expression, which is why this feels obvious now and will not later.
Concept
A relation is any pairing of input values with output values. The inputs form the domain and the outputs form the range. A function is the special kind of relation in which each input is paired with exactly one output.
relation — A pairing of input values with output values. The set of inputs is the domain; the set of outputs is the range.
The four representations carry identical information. Which one you use is a matter of what you are trying to see: mapping diagrams make the function test obvious, and graphs make the shape obvious.
Figure (svg): One relation shown four ways: as ordered pairs, as a table, as a graph, and as a mapping diagram
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 72-72
Section
Section 1
Concept
Ordered pairs, a table, a graph and a mapping diagram all describe the same pairing. Converting between them costs nothing and often makes a question trivial that looked hard in the other form.
domain and range — The domain is the set of all input values of a relation; the range is the set of all output values.
The domain is read off the first coordinates and the range off the second. Repeated values are listed once, because a set records what appears, not how often.
Figure (svg): One relation shown four ways: as ordered pairs, as a table, as a graph, and as a mapping diagram
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 72-72 — Representing Relations
Picture it
Nothing is added or lost moving between them.
Figure (svg): One relation shown four ways: as ordered pairs, as a table, as a graph, and as a mapping diagram
Notice the mapping diagram: the input negative two has two arrows leaving it. That is invisible in the list of ordered pairs and unmistakable here, which is why the diagram is the right tool for the function test.
Worked example
Example 1. Five ordered pairs, and every representation built from them.
\[ \text{For } (-2,-3), (-1,1), (1,3), (2,-2), (3,1): \text{ find the domain and range.} \]
Collect the first coordinates
Why: The first coordinate of each pair is an input, so together they are the domain.
\[ -2, -1, 1, 2, 3 \]
Collect the second coordinates
Why: The second coordinates are the outputs, so together they are the range.
\[ -3, 1, 3, -2, 1 \]
List each value once, in order
Why: A set records which values appear, not how many times. The output 1 appears twice but is listed once.
Plot the five points and draw the mapping diagram
Why: The graph shows the shape; the diagram shows the pairing.
Figure (svg): The solution to Worked example domain, range, and two pictures shown as a ladder of expressions, one row per algebraic move
\[ \text{domain: } \{-2, -1, 1, 2, 3\} \qquad \text{range: } \{-3, -2, 1, 3\} \]
Verify: count the pairs against the mapping diagram
Why: There are five ordered pairs and five arrows in the diagram, so nothing was dropped. The range has only four members because 1 appears as an output twice, which is exactly what the two arrows arriving at 1 show.
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 72-72
Matching
All four carry the same information, but not equally visibly.
Match the pairs
Why: Choosing the right representation is most of the work in these questions. A function test is fastest from a mapping diagram or a graph; a rate-of-change question is fastest from a table; and an exam answer is usually written as ordered pairs because they are compact.
Worked example
Guided Practice 1. Look carefully at the input negative two.
\[ \text{For } (-4,3), (-2,1), (0,3), (1,-2), (-2,-4): \text{ find the domain and range.} \]
Collect the first coordinates
Why: Negative four, negative two, zero, one, and negative two again.
\[ -4, -2, 0, 1, -2 \]
List the domain with each value once
Why: Negative two appears as an input twice, so it is listed once.
Collect the second coordinates and list the range
Why: Three appears twice as an output, so it too is listed once.
Notice what the repeated INPUT means
Why: The input negative two is paired with both 1 and negative 4, which is the thing that will decide the next section's question.
\[ -2\text{ goes to } 1\text{ and to } -4 \]
Figure (svg): The solution to Worked example a relation with a repeated input shown as a ladder of expressions, one row per algebraic move
\[ \text{domain: } \{-4, -2, 0, 1\} \qquad \text{range: } \{-4, -2, 1, 3\} \]
Verify: rebuild the pairs from the two sets
Why: The two sets alone cannot rebuild the relation — many different pairings share these sets — which is a useful reminder that domain and range summarise a relation rather than define it. Counting arrows in the mapping diagram gives five, matching the five original pairs.
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 73-73
Trap
\[ (-4,3), \; (-2,1), \; (0,3), \; (1,-2), \; (-2,-4) \]
Write the range by copying every second coordinate
Why: The list is copied rather than the set being formed.
\[ \text{range: } \{3, 1, 3, -2, -4\} \]
The value 3 appears twice, which a set never does.
\[ (-4,3), \; (-2,1), \; (0,3), \; (1,-2), \; (-2,-4) \]
Collect the second coordinates, then list each distinct value once
Why: Domain and range are sets: they record which values occur, not how often.
\[ \text{range: } \{-4, -2, 1, 3\} \]
Repetition is not an error in the relation — the relation genuinely sends two inputs to 3. It is only the listing that must not repeat.
Sorting
For the relation given by the pairs in Example 1.
Sort into buckets
Sort each value by which set it belongs to. Some belong to both.
Fill the middle
A relation is given by four ordered pairs.
Fill in the blanks
(-2, 2), \; (-2, -2), \; (0, 1), \; (3, 1) \;\Longrightarrow\; \text-2, 0, 3 = \___\}
Why: The first coordinates are negative two, negative two, zero and three, and listing the distinct ones gives three values rather than four. The repetition matters for the function question but not for the domain: negative two is one input, even though it appears in two pairs.
Translation
The same four pairs, in the other notation.
Match the pairs
Why: A table has one row per ordered pair, which means an input can appear on two rows. That is exactly what the first two rows do here, and it is the visual signal that this relation will fail the function test. In a table the tell is a repeated x with different y values.
Section
Section 2
Concept
A function is a relation in which each input is paired with exactly one output. If any input has more than one output, the relation is not a function. Outputs may repeat as much as they like — the restriction is on inputs only.
function — A relation for which each input has exactly one output. If any input of a relation has more than one output, the relation is not a function.
The textbook flags the common misreading directly: a relation may map several different inputs onto the same output and still be a function.
Figure (svg): Two mapping diagrams side by side: one where every input has a single output, and one where an input has two outputs
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 73-73 — the definition and its Avoid Errors note
Picture it
Inputs are restricted; outputs are not.
Figure (svg): Two mapping diagrams side by side: one where every input has a single output, and one where an input has two outputs
A function may be many-to-one but never one-to-many. Every constant function is a spectacular example: every input goes to the same single output, and it is still perfectly a function.
Worked example
Example 2. The test is one question asked of each input.
\[ \text{Diagram A: each of four inputs has one arrow. Diagram B: the input } 1 \text{ has arrows to } -1 \text{ and } 2. \]
Ask the question of every input in diagram A
Why: Each input has exactly one arrow leaving it, so no input is ambiguous.
Conclude for diagram A
Why: The definition is satisfied, so it is a function.
Ask the question of every input in diagram B
Why: One input, namely 1, has two arrows leaving it and therefore two outputs.
\[ B:\text{ input } 1\text{ has two} \]
Conclude for diagram B
Why: A single failing input is enough to disqualify the whole relation.
Figure (svg): The solution to Worked example read two mapping diagrams shown as a ladder of expressions, one row per algebraic move
\[ \text{A: function} \qquad \text{B: not a function} \]
Verify: check that repeated OUTPUTS were not counted against A
Why: If diagram A sends two different inputs to the same output, that is still a function — the definition says nothing about outputs being distinct. Confirming that no input in A has two arrows, rather than that no output has two arrivals, is what makes the check correct.
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 73-73
Definition probe
Only one clause exists: every input has exactly one output.
Sort into buckets
Sort each relation by whether it is a function.
Worked example
Guided Practice 2. The answer surprises people.
\[ \text{Inputs } -2, -1, 0, 1, 3 \text{ all map to the output } -4. \text{ Is it a function?} \]
Ask the question of each input in turn
Why: Negative two has one output; negative one has one output; and so on for all five.
Notice what is repeated and what is not
Why: The OUTPUT negative four is repeated five times, but no input is repeated at all.
Apply the definition literally
Why: The definition restricts inputs only, and every input here has exactly one output.
Conclude
Why: It is a function — a constant one, but a function.
Figure (svg): The solution to Worked example a table where every output is the same shown as a ladder of expressions, one row per algebraic move
\[ \text{a function (a constant function)} \]
Verify: draw the graph and check it with a vertical line
Why: The five points all sit at height negative four, forming part of a horizontal line. Any vertical line meets a horizontal line exactly once, so the graph confirms what the definition said. A horizontal line is a function; a vertical line is not.
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 73-73
Trap
\[ (-2,-4), \; (-1,-4), \; (0,-4), \; (1,-4), \; (3,-4) \]
Reject it because the output -4 appears five times
Why: The definition is applied to outputs instead of inputs.
This would make every constant function a non-function, and horizontal lines would be disqualified from graphing.
\[ (-2,-4), \; (-1,-4), \; (0,-4), \; (1,-4), \; (3,-4) \]
Apply the definition to inputs only
Why: Each input has exactly one output, which is all the definition requires.
Repeated outputs are fine. A function may be many-to-one; it may never be one-to-many.
\[ f(x) = -4 \quad \text{for every } x \]
Two truths and a lie
All three are about the definition of a function.
Eliminate the wrong options
Two of these are true. Knock those out and keep the false one.
Survives elimination: C
Why: The survivor is the false one. A constant function has a domain of any size and a range with exactly one member, which breaks the claim immediately. Domain and range are separate sets and their sizes are unrelated, except that the range can never be larger than the domain for a function.
Counterexample
A classmate proposes a shortcut for spotting functions.
\[ \text{if the ordered pairs all have different } y \text{ values, it is a function} \]
Discussion prompt
Find a relation with all-different y values that is nevertheless not a function, then say what the shortcut has confused with what.
Hint: Keep the outputs distinct and repeat an input.
Answer:
\[ (1, 2), \; (1, 5), \; (3, 9) \]
All three outputs differ, and yet the input 1 has two outputs, so this is not a function. The shortcut has confused the condition on inputs with a condition on outputs.
Distinct outputs describe a different and stricter property — a one-to-one function — which Lesson 6.4 will need when it comes to inverses. It is neither necessary nor sufficient for being a function.
Explain it to yourself
The definition is deliberately one-sided.
\[ \text{each input has exactly one output} \]
Discussion prompt
Explain in your own words why it would be useless to have a rule where one input could give two answers. Then say what practical thing you would be unable to do with such a rule.
Hint: Think about what you do with a function once you have one.
Answer:
The point of a rule is to answer a question: given this input, what is the output? If an input has two outputs the rule refuses to answer, and there is no way to choose between them.
Concretely, you could not graph it as a curve you can read, you could not tabulate it, and you could not substitute into it and get a number. Every use of a function assumes the answer is unique, which is why uniqueness is built into the definition rather than checked later.
Section
Section 3
Concept
A relation is a function if and only if no vertical line meets its graph more than once. That is not a new rule — a vertical line collects every point sharing one x value, so two intersections is exactly one input with two outputs.
vertical line test — A graph represents a function if and only if no vertical line intersects it at more than one point.
The if and only if matters: passing the test proves it is a function, and failing proves it is not. Very few tests in this book run both ways.
Figure (svg): Two graphs with vertical lines drawn through them: one crossing the curve once, one crossing it twice
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 73-73 — Vertical Line Test
Picture it
The same dashed line, applied to two different graphs.
Figure (svg): Two graphs with vertical lines drawn through them: one crossing the curve once, one crossing it twice
Sweeping an imaginary vertical line across the whole graph is the practical version. One place where it meets twice is enough to fail.
Worked example
Example 3. Points per game against age, first for a whole team and then for one player.
\[ \text{Team graph: two players are both aged 28 with different averages. Player graph: one point per season.} \]
Sweep a vertical line across the team graph
Why: At x equal to 28 and again at 29 the line meets two points, because two different players share an age.
\[ \text{two points at } x = 28 \]
Conclude for the team graph
Why: One input, age 28, has two outputs, so the relation is not a function.
Sweep a vertical line across the single-player graph
Why: Each season contributes exactly one point, and no two seasons share an age.
Conclude for the player graph
Why: No vertical line meets it twice, so it is a function.
Figure (svg): The solution to Worked example two scatter graphs shown as a ladder of expressions, one row per algebraic move
\[ \text{team graph: not a function} \qquad \text{player graph: a function} \]
Verify: state what the input actually is in each case
Why: In both graphs the input is age. For the team that is a poor choice of input, because age does not determine a player — several people share one. For a single player it is a good one, because he has exactly one age each season. The test outcome follows from the choice of input, not from the data being messy.
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 74-74
Sorting
Decide from the shape alone, without any equation.
Sort into buckets
Sort each graph by whether it passes the vertical line test.
The sideways parabola is worth remembering: it is the graph of x equals y squared, and it is why Chapter 9 treats parabolas as relations rather than as functions.
Worked example
Guided Practice 3. A tenth season is added with the same average as the ninth.
\[ \text{The player averaged } 24.2 \text{ points in season 10, the same as season 9. Still a function?} \]
Identify what the new point repeats
Why: It repeats an OUTPUT, the average of 24.2 points, not an input.
Check the input of the new point
Why: Season ten means a different age from season nine, so the new point has a new x.
Sweep a vertical line again
Why: No vertical line meets two points, since every age still appears once.
Conclude
Why: Repeating an output never breaks the function property.
Figure (svg): The solution to Worked example adding one point shown as a ladder of expressions, one row per algebraic move
\[ \text{still a function} \]
Verify: contrast with what WOULD have broken it
Why: Had he played two seasons at the same age — which can happen if a season spans his birthday — the graph would have two points at one x and would fail. The difference is entirely whether the repeated coordinate is the input or the output.
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 74-74
Error analysis
A student tests a graph and reaches the wrong conclusion.
Annotate
On: \( \text{a horizontal line at } y = 3 \;\Longrightarrow\; \text{not a function, because a HORIZONTAL line meets it everywhere} \)
Vertical lines test whether it is a function. Horizontal lines test whether the function can be inverted. Two different questions, two different lines.
Prediction
Commit before reasoning.
Predict first
A graph plots each student's height against their shoe size for a class of 30. Is it a function?
Correct: Almost certainly not — two students sharing a shoe size gives one input with two outputs.
This is the same structure as the team scatter graph in Example 3. Whether a relation is a function often depends on what you chose as the input, which is why identifying the input is the first step of the test.
Why: The input here is shoe size, not the student, and in a class of thirty several students will share a size while differing in height. That is one input with two outputs, so a vertical line at that size meets the graph twice. The fix is to change what the input is: height as a function of a named student would be a function, because each student has exactly one height.
Explain it
A classmate has memorised the vertical line test but does not know why it works.
Discussion prompt
In three sentences, explain what all the points on a single vertical line have in common, and why two of them being on the graph means the relation fails the definition.
Hint: Ask what coordinate every point on a vertical line shares.
Answer:
Every point on one vertical line has the same x coordinate — that is what makes the line vertical. So two points of the graph on that line are two outputs paired with a single input.
That is precisely what the definition forbids, which is why the test is not a separate rule but the definition drawn as a picture.
Edge cases
One shape sits exactly on the boundary of the test.
Discussion prompt
What happens when you apply the vertical line test to a vertical line itself? Explain what the domain of that relation is and why the failure is total rather than partial.
Hint: Ask how many outputs the single input has.
Answer:
A vertical line at x equal to 3 has a domain consisting of the single value 3, and that one input is paired with every real number as an output.
So it fails as badly as a relation can: not one input with two outputs, but one input with infinitely many. This is also why a vertical line has no slope in the next lesson — the run is zero, and division by zero is undefined for the same underlying reason.
Section
Section 4
Concept
An ordered pair is a solution of an equation in two variables when substituting both coordinates makes the equation true. The graph of the equation is every such point, which is why a point either lies on the graph or does not — there is no partial membership.
independent and dependent variable — In an equation such as y equals three x minus five, x is the independent variable, the input; y is the dependent variable, because its value depends on the input.
\[ y = 3x - 5 \quad \text{has solution } (2, 1) \text{ since } 1 = 3(2) - 5 \]
Graphing is three steps: build a table, plot enough points to see the pattern, and join them.
Figure (svg): A table of values for y equals negative two x minus one, with the five points plotted and joined by a line
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 74-75 — Graphing Equations in Two Variables
Picture it
Example 4: the equation y equals negative two x minus one.
Figure (svg): A table of values for y equals negative two x minus one, with the five points plotted and joined by a line
Five points were enough to see the pattern. The line then claims that every one of its infinitely many points is also a solution — which is a much bigger claim than the table made, and one worth testing.
Worked example
Example 4, all three steps.
\[ \text{Graph } y = -2x - 1. \]
Choose convenient inputs and compute the outputs
Why: Small integers either side of zero keep the arithmetic clean and the picture centred.
\[ x = -2, -1, 0, 1, 2 \]
Fill in the outputs
Why: At negative two: negative two times negative two is four, minus one is three. Continue across.
\[ y = 3, 1, -1, -3, -5 \]
Plot the five points
Why: They all lie on a straight line, which is the pattern the table was built to reveal.
Join them with a line
Why: The line asserts that every point on it satisfies the equation, not just the five plotted.
Figure (svg): The solution to Worked example graph from a table shown as a ladder of expressions, one row per algebraic move
\[ y = -2x - 1 \]
Verify: test a point that was NOT in the table
Why: Take x equal to 3, which the line puts at y equal to negative seven. Substituting: negative two times three minus one is negative seven — correct. Testing an unplotted point is what justifies drawing the whole line rather than only the five dots.
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 75-75
Prediction
Commit before substituting.
Predict first
Is the point (2, 1) a solution of y equals 3x minus 5?
Correct: Yes — substituting gives 1 equal to 3 times 2 minus 5, which is true.
\[ 1 \overset{?}{=} 3(2) - 5 = 6 - 5 = 1 \quad \checkmark \]
Why: A point is a solution when substituting both coordinates makes the equation true, and there is nothing more to it. Three times two is six, minus five is one, which matches the y coordinate. Graphing is never necessary to answer this kind of question, and substituting is both faster and exact where reading a graph is approximate.
Worked example
Guided Practice 4. Same three steps, and the pattern in the table looks different.
\[ \text{Graph } y = 3x - 2. \]
Build the table with the same inputs
Why: Using the same five inputs makes the comparison with the previous example direct.
\[ x = -2, -1, 0, 1, 2 \]
Compute the outputs
Why: Three times negative two minus two is negative eight; then negative five, negative two, one, four.
\[ y = -8, -5, -2, 1, 4 \]
Notice the direction of the pattern
Why: The outputs INCREASE as the inputs increase, where the previous example's decreased.
\[ \text{outputs rise by } 3\text{ each step} \]
Plot and join
Why: The points are collinear again, and the line rises from left to right.
Figure (svg): The solution to Worked example a line with a positive slope shown as a ladder of expressions, one row per algebraic move
\[ y = 3x - 2 \]
Verify: check the point where the line crosses the vertical axis
Why: At x equal to zero the equation gives y equal to negative two, and the table agrees. That crossing point is the constant term of the equation, a fact Lesson 2.3 will make into a rule — but it can be verified here from the table alone.
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 75-75
Trap
\[ \text{Graph } y = x^2 - 1 \text{ from the table } x = -2, -1, 0. \]
Plot three points and join them with a straight line
Why: Three points are enough for a line, so a line is drawn without asking whether a line is right.
\[ (-2, 3), \; (-1, 0), \; (0, -1) \]
The output differences are -3 then -1, which are not equal — so these three points are not collinear and the straight line is a fiction.
\[ \text{Graph } y = x^2 - 1 \text{ from the table } x = -2, -1, 0, 1, 2. \]
Plot ENOUGH points to recognise the pattern, then join
Why: Step two of the procedure says enough points, and enough means enough to see the shape rather than the smallest number that can be joined.
\[ (-2,3), \; (-1,0), \; (0,-1), \; (1,0), \; (2,3) \]
Five points show a symmetric curve, not a line. Checking the output differences before plotting would have said the same thing: unequal differences mean the graph is not straight.
Pattern
The table of values for y equals negative two x minus one.
Step through it
What would the output be at x equal to 5, and what would the constant drop be for y equals 3x minus 2?
At x equal to 5 the output is negative eleven. For y equals 3x minus 2 the constant change is plus three per step, which is why that line rises where this one falls.
Error analysis
A student tabulates y equals negative two x minus one and one row is wrong.
Annotate
On: \( \begin{array}{c|ccccc} x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 3 & 1 & -1 & -3 & -3 \end{array} \)
Scan the differences down the output column before you plot. A broken rhythm is visible in a second and a mis-plotted point is not.
Comparison
Fill the blanks. The comparison is the point.
Comparison matrix
| x | y = -2x - 1 | y = 3x - 2 |
|---|---|---|
| -2 | 3 | -8 |
| -1 | 1 | -5 |
| 0 | -1 | -2 |
| 1 | -3 | 1 |
| 2 | -5 | 4 |
One column falls by 2 per step and the other rises by 3. Both cross the vertical axis at their own constant term. Those two observations are the whole of Lesson 2.3, arrived at from tables.
Section
Section 5
Concept
Renaming y as f of x names the rule and shows its input in one symbol. It is read as the value of f at x, and the brackets never mean multiplication.
linear function — A function that can be written as f of x equals m x plus b, where m and b are constants. Its graph is a line.
\[ y = mx + b \;\Longleftrightarrow\; f(x) = mx + b \]
Letters other than f may name functions — g and h are common — and using different letters is how you keep two functions apart in the same problem.
Figure (svg): A function machine labelled f with x going in and f of x coming out, and the warning that f of x is not multiplication
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 75-75 — the Reading note on f of x
Picture it
The notation records both which rule and which input.
Figure (svg): A function machine labelled f with x going in and f of x coming out, and the warning that f of x is not multiplication
Writing f of negative four is a request: run negative four through the machine called f. It is substitution with a clearer label.
Worked example
Example 5. One function is linear and one is not, and both are evaluated the same way.
\[ \text{For } f(x) = -x^2 - 2x + 7 \text{ and } g(x) = 5x + 8, \text{ classify each and find the value at } x = -4. \]
Classify f
Why: It contains an x squared term, so it cannot be written in the form m x plus b and is not linear.
Evaluate f at negative four
Why: Substitute with brackets: negative four squared is 16, so the first term is negative 16; negative two times negative four is positive 8.
\[ -16 + 8 + 7 = -1 \]
Classify g
Why: It has exactly the form m x plus b, with m equal to 5 and b equal to 8, so it is linear.
Evaluate g at negative four
Why: Five times negative four is negative twenty, plus eight is negative twelve.
\[ -20 + 8 = -12 \]
Figure (svg): The solution to Worked example classify and evaluate shown as a ladder of expressions, one row per algebraic move
\[ f(-4) = -1 \qquad g(-4) = -12 \]
Verify: check the classification against the graphs
Why: The linear one, g, would plot as a straight line, and its constant change confirms it: stepping x up by one always changes g by exactly five. The other does not have a constant change — from x equal to negative four to negative three the value rises by 3, and from negative three to negative two it rises by 1 — which is what non-linear means.
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 75-75
Discrimination
Do not evaluate anything. Decide from the form.
Sort into buckets
Sort each function by whether it is linear.
Worked example
Guided Practice 6. The x term comes second, which does not change anything.
\[ \text{Tell whether } g(x) = -4 - 2x \text{ is linear, then evaluate at } x = -2. \]
Rewrite it in the standard order
Why: Commutative property of addition, from Lesson 1.1: the terms may be written either way round.
\[ g(x) = -2 x - 4 \]
Classify it
Why: Now the form m x plus b is visible, with m equal to negative 2 and b equal to negative 4.
Substitute negative two, with brackets
Why: Negative two times negative two is positive four.
\[ g(-2) = -4 - 2(-2) \]
Simplify
Why: Negative four plus four is zero.
\[ g(-2) = 0 \]
Figure (svg): The solution to Worked example a linear function in disguise shown as a ladder of expressions, one row per algebraic move
\[ \text{linear, and } g(-2) = 0 \]
Verify: evaluate a neighbouring input and check the constant change
Why: At x equal to negative one, g is negative two; at x equal to zero, g is negative four. Each step of one in x changes g by exactly negative two, which is the coefficient — the signature of a linear function, and confirmation that the classification was right.
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 75-75
Error analysis
A student is asked to evaluate f at 3, where f of x equals 5x plus 8.
Annotate
On: \( f(3) = f \cdot 3 = 5x \cdot 3 + 8 \cdot 3 \)
After evaluating a function at a number, no variable should remain. If an x survives, the substitution did not happen.
Fill the middle
Substituting into a non-linear function, where the brackets matter.
Fill in the blanks
f(x) = -x^2 - 2x + 7 \;\Longrightarrow\; f(-4) = -(-4)^2 - 2(-4) + 7 = -1
Why: Both occurrences of x are replaced by negative four in brackets. The square gives positive sixteen, and the leading minus makes that term negative sixteen; the middle term is negative two times negative four, which is positive eight. Negative sixteen plus eight plus seven is negative one. Writing the substitution line in full, as Lesson 1.2 insisted, is what keeps these signs straight.
Matching
Four expressions that look similar and mean different things.
Match the pairs
Why: The third is the only one involving multiplication, and it is written with the 3 OUTSIDE the function name. The fourth changes the input rather than the output, which is the distinction Lesson 2.7 turns into horizontal versus vertical shifts of a graph. For the given rule, f of 3 is 23, three times f of x is 15x plus 24, and f of x plus 3 is 5x plus 23.
Commit first
Answer, then rate your confidence honestly.
Predict first
For a linear function, is the range always all real numbers?
Correct: No — a constant function has a range with exactly one value.
\[ f(x) = 3 \;\Longrightarrow\; \text{range} = \{3\} \qquad f(x) = 2x + 1 \;\Longrightarrow\; \text{range} = \mathbb{R} \]
Why: A linear function is anything of the form m x plus b, and that includes m equal to zero, which gives a constant. Its graph is a horizontal line and its range is a single number. For every other linear function, with m not zero, the range really is all real numbers, because the line keeps rising or falling without bound. The constant case is the sole exception and it is easy to forget.
Comparison
Fill the blanks. Each row is stricter than the one above it.
Comparison matrix
| Term | Requires | Example that fits but not the next row down |
|---|---|---|
| Relation | any pairing of inputs with outputs | (1,2), (1,5) |
| Function | each input has exactly one output | f(x) = x^2 |
| Linear function | the form m x plus b | none - this is the strictest row |
| Vertical line test | no vertical line meets the graph twice | a circle fails it |
Every linear function is a function, and every function is a relation. None of those implications runs backwards, which is what the middle column records.
Pattern
One routine answers every question in this lesson.
Step one is the one that gets skipped. A relation that fails the function test often passes it once you swap which variable is the input.
OpenStax Algebra and Trigonometry 2e, §3.1 Functions and Function Notation §3.1
Check
Domain and range. Watch for repeats.
Check your understanding
For the relation (-4, 3), (-2, 1), (0, 3), (1, -2), (-2, -4), what is the domain?
Answer: A
Why: The domain is the set of first coordinates, listed once each. Negative two appears in two pairs but is a single input, so the domain has four members.
Check
The function test. Apply it to inputs only.
Check your understanding
Which of these relations is NOT a function?
Answer: A
Why: The input negative two is paired with both 1 and negative 4, so one input has two outputs and the definition fails. One such input is enough to disqualify the whole relation.
Check
Function notation and classification.
Check your understanding
For f(x) = -x^2 - 2x + 7, what is f(-4), and is f linear?
Answer: A
Why: Substituting gives -(-4)^2 - 2(-4) + 7, which is -16 + 8 + 7, or -1. The x squared term means f cannot be written as m x plus b, so it is not linear.
Real world
A delivery app records, for each order, the postcode it went to and the minutes it took.
Discussion prompt
Is minutes-taken a function of postcode? Is it a function of order number? Explain both answers in terms of the definition, and say what that tells you about choosing an input when you build a model.
Hint: Ask whether the same input value can occur twice with different outputs.
Answer:
Not a function of postcode: two orders to the same postcode will almost certainly take different times, so one input has two outputs. A vertical line at that postcode would meet the scatter graph twice.
It IS a function of order number, because each order number occurs once and has exactly one duration. Order numbers are unique by construction, which is precisely why databases assign them.
The practical lesson: whether something is a function depends on the input you choose, and a good input is one that identifies a single case. That is the same reasoning that made the team scatter graph fail and the single-player graph pass.
Commit first
Answer, then rate your confidence honestly.
Predict first
Can the range of a function have fewer members than its domain?
Correct: Yes — several inputs may share an output.
\[ (-2,-4), (-1,-4), (0,-4), (1,-4) \;\Longrightarrow\; \text{domain has 4, range has 1} \]
Why: The definition restricts inputs only, so many-to-one is allowed and a constant function takes it to the extreme: any domain at all, and a range of exactly one value. What can never happen is the reverse, a range larger than the domain, because each input contributes at most one output. This asymmetry is the whole content of the definition.
Explain it
They have graphed points but never heard the word function.
Discussion prompt
In four sentences or fewer, explain what a function is, why repeated outputs are fine but repeated inputs are not, and give them the picture test they can run on any graph.
Hint: Lead with the idea of a rule that always gives an answer.
Answer:
A function is a rule that, given any input it accepts, produces exactly one output — so you can always ask it a question and get a single answer. Two different inputs giving the same answer is harmless; one input giving two answers means the rule cannot decide, which makes it useless as a rule.
The picture test: sweep a vertical line across the graph. If it ever touches the graph in two places at once, that x has two answers and it is not a function.
Exit ticket
Name the weakest spot before you close the deck.
Predict first
Which of these would you least want handed to you cold?
Correct: Whichever you picked is tonight's ten minutes, and each has a one-line fix.
Why: For repeats, ask whether the repeated value is an input or an output — only inputs break it. For the vertical line test, sweep the line mentally across the whole graph rather than checking one place. For tables, choose small integers either side of zero and check the output differences are constant. For notation, remember the brackets name an input and never mean multiplication. Do five of your chosen kind rather than twenty mixed ones.
Connect it up
Paper. Fifteen minutes.
Draw it
Across the top of a page write one relation as five ordered pairs, then represent that same relation as a table, a graph and a mapping diagram, so all four sit side by side. Circle the domain in one colour and the range in another. Underneath, write the definition of a function in your own words, and beside it draw one mapping diagram that passes and one that fails, marking on the failing one exactly which input broke it. In the middle of the page graph an equation of your own of the form y equals m x plus b by building a table of five values, and draw a dashed vertical line through your graph to show it passes the test. At the bottom, rewrite your equation in function notation and evaluate it at two inputs, writing the substitution line out in full each time.
If your mapping diagram that fails has two arrows arriving at one output rather than two leaving one input, reread Section 2 — that is the single most common misreading of the definition.
Recap
Five things, and the second is a definition the rest of this book assumes.
| If you see | Ask |
|---|---|
| A repeated value in a relation | Is it a repeated input or a repeated output? |
| A graph | Does any vertical line meet it twice? |
| An equation in two variables | Which variable is the input? |
| f of something | What am I substituting for x? |
| A function to classify | Does m x plus b fit it? |
Lesson 2.2 takes the constant change you noticed in the tables and gives it a name and a formula: slope.
McDougal Littell Algebra 2 (Texas Edition), Ch. 2 Linear Equations and Functions — Lesson 2.1 Represent Relations and Functions §2.1, pp. 72-79 — everything on these slides traces back here
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