Turns the loose word 'function' into a test you can run: every input has exactly one output. Builds function notation as a name for an output rather than a multiplication, separates evaluating f(3) from solving f(x) = 3, and introduces the vertical and horizontal line tests as the same definition seen from two directions.
Subject: Precalculus · 65 slides · symbolic lesson
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Title
Precalculus · Chapter 1 — Functions
§1.1 Functions and Function Notation, pp. 10-40
Objectives
Five things, each one you can check yourself on paper.
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 10-40 — the pages these objectives are drawn from
Warm-up
You have used the word function since Algebra 1. This lesson makes it precise, so it is worth finding out what you currently mean by it.
Discussion prompt
A machine takes a person's name and returns their birthday. A second machine takes a birthday and returns a person's name. Which of these is reliable, and what exactly goes wrong with the other?
Hint: Try it with a room of thirty people. Which direction can get stuck?
Answer:
The first is reliable: each person has exactly one birthday, so the machine always knows what to return.
The second is not. Two people in the room may share a birthday, and then the machine has two equally good answers and no way to pick. It is not that the answer is hard to find — it is that there is no single right answer to find.
That is the whole definition, and it is one-directional. A function may send two different inputs to the same output; it may never send one input to two outputs. Most of the mistakes in this lesson come from applying that rule in the wrong direction.
Concept
A function is a rule that assigns to each input exactly one output. The word 'exactly' does the work: not at most one, which would allow inputs with no answer, and not at least one, which would allow inputs with two.
function — A relation in which each element of the domain is paired with exactly one element of the range. The pairing may send several inputs to the same output, but never one input to several outputs.
\[ f: x \longmapsto f(x), \qquad \text{each } x \text{ in the domain has exactly one } f(x) \]
Notice that the definition is not symmetric. Whether a rule is a function depends entirely on what happens as you leave each input, and says nothing about what arrives at each output. The rule that squares every number is a function even though 3 and negative 3 both land on 9.
Figure (svg): Two mapping diagrams side by side: on the left a relation where every input has exactly one arrow leaving it, labelled a function; on the right a relation where one input has two arrows leaving it, labelled not a function
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 10-12
Section
Section 1
Concept
A relation is any set of ordered pairs. It earns the name function when no input appears twice with different partners.
Students lose more marks on this section to the last two bullets than to anything else. The test is one-directional, and reading it as 'no repeats anywhere' rejects perfectly good functions.
Figure (svg): Two mapping diagrams side by side: on the left a relation where every input has exactly one arrow leaving it, labelled a function; on the right a relation where one input has two arrows leaving it, labelled not a function
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 10-13
Picture it
Everything in this lesson can be recovered from this pair of pictures.
Figure (svg): Two mapping diagrams side by side: on the left a relation where every input has exactly one arrow leaving it, labelled a function; on the right a relation where one input has two arrows leaving it, labelled not a function
Cover the labels and ask only one question of each diagram: does any single circle on the left have two arrows leaving it? That question, and no other, decides it.
Worked example
Check the inputs, and only the inputs.
\[ \text{Is } \{(1,2),\,(3,4),\,(5,4),\,(7,8)\} \text{ a function?} \]
List the inputs
Why: The first coordinate of each pair.
Check for a repeated input
Why: Scan the list for any value appearing twice.
Ignore the repeated output
Why: The output 4 occurs twice, which the definition does not forbid.
Conclude
Why: No input has two partners.
Figure (svg): The solution to Worked example is this set of pairs a function shown as a ladder of expressions, one row per legal move
\[ \text{Yes — the inputs } 1,3,5,7 \text{ are distinct, so each has exactly one output.} \]
Verify: state what would have broken it
Why: Adding the pair with input 3 and output 9 would break it, because 3 would then have both 4 and 9. Adding a pair with input 9 and output 4 would not break it, because a third arrival at 4 is irrelevant.
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 11-11
Sorting
Run one test on each: does any input appear twice with different outputs?
Sort into buckets
Sort each relation.
Worked example
The same test, applied to a table of values rather than a list of pairs.
\[ \begin{array}{c|cccc} x & 2 & 4 & 6 & 4 \\ \hline y & 5 & 7 & 9 & 11 \end{array} \]
Read the top row as the inputs
Why: A table lists the pairs column by column.
Find the repeat
Why: The input 4 appears in column two and again in column four.
\[ 4\text{ appears twice} \]
Compare their outputs
Why: The first time it gives 7, the second time 11.
\[ 7\text{ and } 11\text{ differ} \]
Conclude
Why: One input, two different outputs.
Figure (svg): The solution to Worked example a table that fails shown as a ladder of expressions, one row per legal move
\[ \text{Not a function: } f(4) \text{ would have to be both } 7 \text{ and } 11. \]
Verify: say what would fix it
Why: Changing the last output from 11 to 7 fixes it: the input 4 would then give 7 both times, which is one pair written twice, not two pairs. Changing the last input from 4 to 8 also fixes it, by removing the repeat.
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 12-13
Trap
\[ \{(1,4),\,(2,4),\,(3,4)\} \]
Reject it because the output 4 repeats
Why: Three pairs share the same second coordinate, which looks like a collision.
The conclusion drawn is that this cannot be a function because the value 4 is used three times.
It is a function. The inputs are 1, 2 and 3, all different, so each has exactly one output. That those outputs coincide is not the definition's concern.
This is the constant rule that returns 4 no matter what goes in — a perfectly good function, and one used constantly. A horizontal line passes the vertical line test at every point.
Repeated outputs cost nothing; repeated inputs with different outputs cost everything. When the horizontal-line idea arrives later in this lesson, it will finally give repeated outputs something to mean — but it answers a different question, not this one.
Prediction
A relation pairs each student in a school with the class they are sitting in right now.
Predict first
Is it a function, and does the answer depend on class sizes?
Correct: A function, and class sizes are irrelevant.
Why: The inputs are students and the outputs are classes. Each student is sitting in exactly one class right now, so every input has exactly one output, which is the entire requirement. Many students sharing a class means many inputs share an output, and the definition never mentions that. Reversing the relation, pairing each class with its students, is what would fail.
Two truths and a lie
Two statements about the definition are true and one is false. Rule out the true ones.
Eliminate the wrong options
One of these claims about the definition is wrong.
Survives elimination: B
Why: The false claim is B, and it names the definition's only prohibition. Listing both outputs does not legitimise them — it is exactly what makes the relation fail the test. The other two are genuinely true, and the first of them is the one students most often reject by mistake.
Socratic
The definition could have been written more loosely.
Discussion prompt
What would go wrong if a function were allowed to assign at least one output to each input rather than exactly one?
Hint: Think about what you want to be able to write down once a function has a name.
Answer:
You would lose the right to write f(3) at all. The notation names a single number; if the input 3 had two outputs, the symbol f(3) would be ambiguous and could not be used in an equation.
Every later technique depends on that. Composition in Section 1.4 feeds one output into the next rule and needs to know which. Inverses in Section 1.7 need a single output to send back. Limits and derivatives in Chapter 12 track one output as the input moves.
So 'exactly one' is not fussiness. It is the condition that makes the notation, and everything built on it, well defined — which is why the word is worth the emphasis it gets.
Section
Section 2
Concept
The symbol f(x) is read 'f of x' and names the output that the rule f produces from the input x. The parentheses signal an input, not a product.
Because the parentheses look like multiplication, f(a plus b) is very often expanded as if it were f times a plus f times b. It is not, and no rule of algebra says it is. The parentheses here are punctuation, marking where the input goes.
Figure (svg): A function drawn as a machine: an input x enters on the left, a rule box in the middle reads two x plus one, and the output two x plus one leaves on the right
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 14-17
Picture it
Reading the notation as a machine makes the substitution rule obvious rather than memorised.
Figure (svg): A function drawn as a machine: an input x enters on the left, a rule box in the middle reads two x plus one, and the output two x plus one leaves on the right
The slot takes whatever you hand it. If you hand it the expression x plus 1, then every x in the rule becomes the whole expression x plus 1, brackets and all.
Worked example
The one place this goes wrong is a dropped bracket.
\[ \text{Given } f(x) = x^2 - 4x, \text{ find } f(-3). \]
Rewrite the rule with a hole where x was
Why: Substitution is easier to get right when the slot is visible.
\[ f() = () ^{2} - 4() \]
Drop negative 3 into every hole, keeping the brackets
Why: The brackets are what make the square apply to the sign as well.
\[ (-3) ^{2} - 4(-3) \]
Evaluate the square first
Why: A negative squared is positive.
\[ 9 - 4(-3) \]
Finish the arithmetic
Why: Subtracting a negative adds.
\[ 9 + 12 = 21 \]
Figure (svg): The substitution of the input negative three into the rule f of x equals x squared minus four x, shown as three stages: the rule with a hollow box in place of x, the box filled with negative three in brackets, and the arithmetic evaluated to twenty one
\[ f(-3) = (-3)^2 - 4(-3) = 9 + 12 = 21 \]
Verify: check the sign trap deliberately
Why: Without the brackets the first term reads as the negative of three squared, which is negative 9, and the answer would come out as 3 instead of 21. Since both terms of the rule are being fed a negative number, both change, and the true answer is comfortably positive.
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 17-18
Notation
Each part of this expression has a job.
Annotate
On: \( g(t) = 3t - 5, \qquad g(4) = 7 \)
Nothing in the line involves multiplying g by anything. The parentheses mark the input slot, and that is their only job here.
Worked example
The slot accepts an expression exactly as it accepts a number.
\[ \text{Given } f(x) = x^2 - 4x, \text{ find } f(a+1). \]
Substitute the whole expression, in brackets
Why: Every occurrence of x becomes the same bracketed expression.
\[ (a + 1) ^{2} - 4(a + 1) \]
Expand the square
Why: Use the square of a binomial; do not square term by term.
\[ a ^{2} + 2 a + 1 - 4(a + 1) \]
Distribute the negative four
Why: Both terms inside the bracket are multiplied.
\[ a ^{2} + 2 a + 1 - 4 a - 4 \]
Collect like terms
Why: Combine the a terms and the constants.
\[ a ^{2} - 2 a - 3 \]
Figure (svg): The solution to Worked example evaluate at an expression shown as a ladder of expressions, one row per legal move
\[ f(a+1) = (a+1)^2 - 4(a+1) = a^2 - 2a - 3 \]
Verify: test it at one value
Why: Putting a equal to 2 in the answer gives 4 minus 4 minus 3, which is negative 3. Going the other way, a equal to 2 means the input was 3, and the original rule at 3 gives 9 minus 12, which is also negative 3. The two agree.
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 18-19
Error analysis
A student is asked for f(a plus b) given that f is the squaring rule.
Annotate
On: \( f(x) = x^2 \;\Longrightarrow\; f(a+b) = f(a) + f(b) = a^2 + b^2 \)
A function name is not a quantity, so nothing may be distributed over it or cancelled against it. When in doubt, test the claim on small numbers — it takes ten seconds and settles the question.
Matching
Notation is compressed English. Uncompress it.
Match the pairs
Why: The first is a completed statement of fact about one input. The second is an equation to solve, because the input is the unknown. The third is unusual but legal: it defines x to be a particular output. The fourth is the sentence that lets graphs and function notation be used interchangeably, and it is why the vertical axis of a graph can be labelled either y or f of x.
Faded example
Given h(x) equal to x squared plus 2x, evaluate at negative two.
Fill in the blanks
h(-2) = (-2)^2 + 2(-) = 4 0 4 = ___
Why: Negative two goes into both slots with its brackets intact. The square gives positive 4 and the second term gives negative 4, so they cancel and the output is 0. Without brackets the first term would have read as the negative of two squared and the answer would have been negative 8, which is the standard version of this mistake.
Counterexample
A classmate claims that f(2x) is always the same as 2f(x).
Discussion prompt
Find a function for which this fails, and then a function for which it happens to hold.
Hint: Try squaring, and then try a rule that only multiplies.
Answer:
It fails for squaring. With the rule that squares its input, f(2x) is four x squared while 2f(x) is only two x squared. Those agree only when x is zero.
It holds for the tripling rule. With f(x) equal to 3x, f(2x) is 6x and 2f(x) is also 6x, for every x. So the claim is not always false — it is just not a rule.
That is the point worth taking away: a statement that holds for some functions is not an identity, and finding one function where it fails is enough to retire it. Rules that survive this test for every function are rare, and Section 1.4 will be careful about which ones do.
Section
Section 3
Concept
A function can be presented as a set of pairs, a table, or a graph. The test never changes; only the way a repeated input shows itself does.
The graph case is the one worth dwelling on. Two points stacked vertically have the same x and different y, which is exactly one input with two outputs. Sweeping a vertical line across the picture is simply a way of checking every input at once.
| presented as | an input is | the failure looks like |
|---|---|---|
| a set of pairs | a first coordinate | the same first coordinate twice, with different partners |
| a table | an entry in the input row | the same entry in the input row, under different outputs |
| a graph | an x value | two plotted points directly above one another |
Figure (svg): Two graphs side by side with a vertical dashed line drawn through each: a parabola opening upward, which the line crosses once, and a sideways parabola, which the line crosses twice
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 20-25
Picture it
A vertical line is the set of all points sharing one x value, so it collects everything the definition cares about at that input.
Figure (svg): Two graphs side by side with a vertical dashed line drawn through each: a parabola opening upward, which the line crosses once, and a sideways parabola, which the line crosses twice
The sideways parabola on the right is a perfectly respectable curve and a perfectly respectable relation. It simply is not a function of x — though it is a function of y, which is a distinction Section 1.7 will use.
Worked example
A circle of radius 3 centred at the origin.
\[ \text{Does } x^2 + y^2 = 9 \text{ define } y \text{ as a function of } x? \]
Pick an input inside the circle
Why: Any x strictly between negative 3 and 3 will do.
\[ \text{take } x = 0 \]
Solve for the outputs at that input
Why: The equation becomes y squared equals 9.
\[ y ^{2} = 9 \]
Count them
Why: A positive number has two square roots.
\[ y = 3\text{ and } y = -3 \]
Conclude
Why: One input with two outputs is exactly the disqualifier.
Figure (svg): The solution to Worked example apply the vertical line test shown as a ladder of expressions, one row per legal move
\[ \text{No. At } x=0, \; y = 3 \text{ and } y = -3 \text{ both satisfy the equation.} \]
Verify: confirm with the picture
Why: A vertical line through the middle of a circle enters at the bottom and leaves at the top, crossing twice. The only vertical lines that meet a circle once are the two tangents at the far left and far right, and one input out of infinitely many is nowhere near enough.
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 23-24
Discrimination
A vertical line meeting a graph zero times and meeting it twice mean different things.
Sort into buckets
Sort each observation by what it tells you.
Worked example
Using the graph on the visual slide, with f(x) equal to negative x squared plus 2x plus 3.
\[ \text{From the graph, find } f(1) \text{ and then solve } f(x) = 0. \]
For f(1), go up from x equal to 1
Why: Evaluation starts at an input on the horizontal axis.
\[ \text{up from } x = 1 \]
Read across to the vertical axis
Why: The height of the curve there is the output.
\[ f(1) = 4 \]
For the equation, travel along the height 0
Why: Solving starts at an output and looks for inputs.
\[ \text{along } y = 0 \]
Collect every crossing
Why: Two points of the curve sit at that height.
\[ x = -1\text{ and } x = 3 \]
Figure (svg): A graph of a function with two readings marked: a vertical arrow up from x equals one to the curve and across to y equals three, showing evaluation, and a horizontal arrow from y equals zero across to two crossings, showing solving
\[ f(1) = 4, \qquad f(x) = 0 \text{ at } x = -1 \text{ and } x = 3 \]
Verify: check against the rule
Why: The rule at 1 gives negative 1 plus 2 plus 3, which is 4. Factoring the rule gives the negative of the product of x plus 1 and x minus 3, which is zero exactly at negative 1 and 3. Graph and algebra agree.
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 25-27
Trap
\[ f(x) = \sqrt{x}, \qquad \text{tested at } x = -4 \]
Draw a vertical line at negative four and observe no crossing
Why: The line meets the curve zero times rather than once.
The conclusion drawn is that the square root rule fails the vertical line test, because a vertical line did not cross it exactly once.
The test is about lines that cross more than once, not lines that miss. A vertical line at negative 4 misses because negative 4 is not in the domain — there is no output there to find.
An input outside the domain is not a counterexample; it is simply not being asked about. The definition says each input in the domain has exactly one output.
Stated carefully: a graph is a function when no vertical line crosses it more than once. Zero crossings tell you about the domain, which is the subject of the next section, and nothing about whether the rule is a function.
Prediction
A graph consists of the single point at the origin and nothing else.
Predict first
Is it a function?
Correct: Yes, on a domain containing only the input 0.
Why: Its domain is the single value 0, and at that one input there is exactly one output, which is 0. Every other vertical line misses, and missing is a statement about the domain rather than a failure. Small and strange domains are still domains, and this is a legitimate, if very short, function.
Sorting
A curve can be a function in one direction and not the other. Sweep both kinds of line.
Sort into buckets
Sort each curve by which sweep it survives.
Explain it to yourself
The vertical line test is often stated as a trick to memorise.
Discussion prompt
Explain, without using the phrase 'line test', why two points stacked vertically disqualify a graph.
Hint: What do two such points have in common, and what do they not?
Answer:
Two points on the same vertical line have the same x coordinate and different y coordinates. Read as input and output, that is one input paired with two outputs.
So the picture is not a separate criterion that happens to work. It is the definition, drawn — the only thing the line does is check every input at once instead of one at a time.
Seeing it that way is what makes the next test, the horizontal one, easy rather than another thing to memorise: swap the roles of the coordinates and you are asking whether two inputs share an output.
Section
Section 4
Concept
Evaluating hands the function an input and asks what comes out. Solving hands it an output and asks which inputs produce it. The notation for the two looks almost identical, and the answers behave completely differently.
This asymmetry is the definition making itself felt. A function may send many inputs to one output, so an equation asking which inputs give a particular output can perfectly well have several answers — and a well-drilled student who expects a single number will stop after finding the first one.
Figure (svg): A two-column contrast: on the left, evaluating a function means you are given x and asked for y; on the right, solving means you are given y and asked for x
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 19-22
Picture it
One graph answers both questions, and the direction you travel decides which one you asked.
Figure (svg): A graph of a function with two readings marked: a vertical arrow up from x equals one to the curve and across to y equals three, showing evaluation, and a horizontal arrow from y equals zero across to two crossings, showing solving
The across-then-down journey can land on two inputs, as it does here. The up-then-across journey never can, and that is the definition at work.
Worked example
The output is given; the inputs are wanted.
\[ \text{Given } f(x) = x^2 - 2x, \text{ solve } f(x) = 3. \]
Set the rule equal to the given output
Why: This is what the notation is asking for.
\[ x ^{2} - 2 x = 3 \]
Move everything to one side
Why: A quadratic is solved against zero.
\[ x ^{2} - 2 x - 3 = 0 \]
Factor
Why: Two numbers multiplying to negative 3 and adding to negative 2.
\[ (x - 3) (x + 1) = 0 \]
Read both roots
Why: A product is zero when either factor is.
\[ x = 3\text{ and } x = -1 \]
Figure (svg): The solution to Worked example solve f x 3 shown as a ladder of expressions, one row per legal move
\[ x = 3 \quad \text{and} \quad x = -1 \]
Verify: substitute both back
Why: At 3 the rule gives 9 minus 6, which is 3. At negative 1 it gives 1 plus 2, which is also 3. Both inputs genuinely produce the output 3, which is allowed — and stopping after the first would have missed half the answer.
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 21-22
Sorting
Decide which task each instruction is asking for before doing any algebra.
Sort into buckets
Sort each instruction.
Worked example
Same function, the other task, so that the contrast is exact.
\[ \text{Given } f(x) = x^2 - 2x, \text{ find } f(3) \text{ and } f(-1). \]
Substitute 3
Why: One input, so one computation.
\[ 3 ^{2} - 2(3) \]
Compute
Why: Nine minus six.
\[ f(3) = 3 \]
Substitute negative 1, in brackets
Why: The square must apply to the sign.
\[ (-1) ^{2} - 2(-1) \]
Compute
Why: One plus two.
\[ f(-1) = 3 \]
Figure (svg): The solution to Worked example the same rule, evaluated shown as a ladder of expressions, one row per legal move
\[ f(3) = 3, \qquad f(-1) = 3 \]
Verify: notice what this shows
Why: Two different inputs produced the same output, which is why the equation in the previous example had two solutions. The two worked examples are two readings of one fact, and the pair of answers here is exactly the pair of roots there.
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 21-22
Error analysis
A student solves an equation and reports one answer.
Annotate
On: \( f(x) = x^2, \; f(x) = 25 \;\Longrightarrow\; x = 5 \)
Evaluation gives one answer because the definition promises it. Solving carries no such promise, so the number of answers has to be found rather than assumed.
Prediction
The graph of a function rises, turns over, and falls, reaching a highest point of 10.
Predict first
How many solutions does the equation setting the output to 10 have?
Correct: Exactly one, at the turning point.
Why: The height 10 is reached only at the very top. A horizontal line drawn at that height touches the curve at the single turning point rather than cutting through it, so there is one input. At any height slightly below 10 the same line would cut twice, giving two inputs, and above 10 it would miss and give none — which is a good illustration that the number of solutions depends on the output you ask about.
Fill the middle
Solving the equation that sets the tripling-and-shifting rule to 11.
Fill in the blanks
3x - 4 = 11 \;\Longrightarrow\; 3x = 15 \;\Longrightarrow\; x = 5
Why: Adding 4 to both sides gives 15, and dividing by 3 gives 5. Checking: the rule at 5 gives 15 minus 4, which is 11, as required. This equation has exactly one solution because the rule is linear with a nonzero slope, so no horizontal line can cut it twice — a fact the next section makes systematic.
Missing information
You are told only that f(2) equals 9.
Discussion prompt
Can you find every input whose output is 9? If not, what is the smallest extra thing you would need?
Hint: Does knowing one arrow tell you about any other?
Answer:
No. Knowing f(2) equals 9 tells you that 2 is one such input. It says nothing about whether any other input also lands on 9, because a function is permitted to send several inputs to the same output.
The smallest sufficient extra fact is that f is one-to-one, which is precisely the promise that no other input shares an output. With that, 2 is the complete answer.
Failing that, you would need the rule itself, or its graph, to hunt for other crossings at that height. This is the exact gap the last section of this lesson fills.
Section
Section 5
Concept
A function is one-to-one when different inputs always produce different outputs. Every function forbids one input with two outputs; a one-to-one function additionally forbids two inputs with one output.
one-to-one function — A function in which no two distinct inputs share an output. Equivalently, no horizontal line meets its graph more than once. This is exactly the condition under which the function can be reversed.
The reason to care arrives in Section 1.7. To undo a function you must be able to look at an output and name the input it came from, and that is possible exactly when no two inputs share an output. So this test is the entry requirement for having an inverse.
Figure (svg): Two graphs with a horizontal dashed line across each: a cubic, which the line meets once, and a parabola, which the line meets twice, illustrating the horizontal line test for one-to-one functions
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 28-33
Picture it
The two tests differ in one word and answer entirely different questions.
Figure (svg): A summary card contrasting the vertical line test and the horizontal line test, giving what each one asks, what a failure means, and which later section needs it
Failing the vertical test means the picture was never a function. Failing the horizontal test means it is a function that cannot be reversed. Only the first is a disqualification.
Worked example
Two rules that look similar and differ on this test.
\[ \text{Are } f(x) = x^2 \text{ and } g(x) = x^3 \text{ one-to-one?} \]
Look for two inputs sharing an output under f
Why: Try a number and its negative.
\[ f(3) = f(-3) = 9 \]
Conclude for f
Why: Two distinct inputs, one output.
Try the same attack on g
Why: Cubing preserves sign, so a negative cannot match a positive.
\[ g(3) = 27, g(-3) = -27 \]
Confirm for g in general
Why: The cubing rule is increasing everywhere, so it never revisits a value.
Figure (svg): Two graphs with a horizontal dashed line across each: a cubic, which the line meets once, and a parabola, which the line meets twice, illustrating the horizontal line test for one-to-one functions
\[ f(x) = x^2 \text{ is not one-to-one}; \quad g(x) = x^3 \text{ is.} \]
Verify: confirm with horizontal lines
Why: The horizontal line at height 9 cuts the parabola at both 3 and negative 3, so the parabola fails. Any horizontal line cuts the cubic exactly once, because the cubic rises without ever turning back, so it passes.
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 29-30
Sorting
Every one of these is a function. The question is whether it is one-to-one.
Sort into buckets
Sort each rule.
Worked example
A rule that fails the test can often be repaired by throwing inputs away.
\[ \text{Make } f(x) = x^2 \text{ one-to-one by restricting its domain.} \]
Identify why it fails
Why: Each positive output is reached from both sides of zero.
\[ 3\text{ and } -3\text{ both give } 9 \]
Keep only one side
Why: Discard the inputs below zero.
Check the test on what remains
Why: On that half, the rule only increases.
Note the outputs are unchanged
Why: Every nonnegative output is still produced, just once now.
\[ \text{range still } y \ge 0 \]
Figure (svg): The solution to Worked example restricting the domain to rescue it shown as a ladder of expressions, one row per legal move
\[ f(x) = x^2 \text{ on } x \ge 0 \text{ is one-to-one.} \]
Verify: check nothing was lost
Why: Every output the original rule produced is still produced: the value 9 still comes from 3. What was discarded is the duplicate route to it. That is why Section 1.7 can define the square root as the inverse of the squaring rule, but only after this restriction is agreed.
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 31-32
Trap
\[ f(x) = x^2, \qquad f(3) = f(-3) = 9 \]
Observe that two inputs give the same output
Why: The output 9 is produced twice, which looks like a collision.
The conclusion drawn is that the squaring rule is therefore not a function.
The squaring rule is a function. Every input has exactly one output: 3 gives 9 and only 9; negative 3 gives 9 and only 9. Nothing is ambiguous.
What has been found is that it is not one-to-one, which is a different and much weaker statement. It remains a function in perfectly good standing.
The practical difference: a non-function cannot be used at all, while a function that is not one-to-one is used constantly and merely cannot be inverted without first restricting its domain. Keep the two failures apart — one is fatal and the other is a footnote.
Elimination
A function has failed the horizontal line test. Rule out the conclusions that genuinely follow.
Eliminate the wrong options
One of these conclusions does NOT follow.
Survives elimination: C
Why: C is the conclusion that does not follow, and it is the survivor. Only the vertical line test can disqualify a graph from being a function; the horizontal test asks a different question entirely, and failing it leaves the rule a function in perfectly good standing that merely cannot be reversed as it stands.
Ranking
Each condition in this list implies the ones weaker than it.
Put in order
Why: A relation is any set of pairs and asks nothing. A function is a relation that additionally forbids one input with two outputs. A one-to-one function is a function that additionally forbids two inputs with one output. Each step adds a restriction, so the collections shrink: every one-to-one function is a function, and every function is a relation, while neither converse holds.
Edge cases
The constant rule sends every input to 7.
Discussion prompt
It is a function. How badly does it fail the one-to-one test, and is there any domain restriction that repairs it?
Hint: How many inputs would have to survive the restriction?
Answer:
It fails as badly as possible: every pair of distinct inputs shares an output. The horizontal line at height 7 does not cross the graph at a few points, it lies along the whole of it.
A restriction does repair it, but only a brutal one: keep exactly one input. With a single input there are no two distinct inputs to collide, so the condition holds vacuously.
That extreme case is worth seeing because it shows the restriction trick from the previous worked example is not always useful. Repairing the squaring rule cost half its domain and kept all of its outputs; repairing a constant rule costs everything and keeps one point.
Comparison
Fill the blanks from memory. These four are routinely confused with one another, and laid out this way the differences are small and total.
Comparison matrix
| what is given | what is wanted | how many answers | |
|---|---|---|---|
| evaluating f(3) | an input | the output | exactly one, always |
| solving f(x) = 3 | an output | every input giving it | zero, one, or many |
| vertical line test | a graph | is it a function | yes or no |
| horizontal line test | a graph of a function | is it one-to-one | yes or no |
The first two rows are the same function read in opposite directions. The last two rows are the same sweep applied along opposite axes. Nothing here is a separate technique to memorise.
Pattern
The same five steps work whether you are handed a set of pairs, a table, a graph, or an equation.
Step 3 is where most marks are lost, in both directions: rejecting a function because an output repeated, and accepting a non-function because the repeated input was in a different column of the table and went unnoticed.
OpenStax Algebra and Trigonometry 2e, §3.1 Functions and Function Notation §3.1
Check
Run the test on the inputs, and only the inputs.
Check your understanding
Which of these sets of ordered pairs is a function?
Answer: A
Why: In the first set the inputs are 1, 2, 3 and 4, all distinct, so each has exactly one output and it is a function. That the outputs happen to be double the inputs is a bonus, not part of the test.
Check
Keep the brackets around the input.
Check your understanding
Given f(x) = x squared minus 3x, what is f(-2)?
Answer: A
Why: Substituting negative 2 with brackets gives the square of negative 2, which is 4, minus 3 times negative 2, which is negative 6. So the expression is 4 plus 6, which is 10. Both terms are affected by the sign of the input, and both push the answer upward.
Check
Two tests, two separate verdicts.
Check your understanding
A graph passes the vertical line test but fails the horizontal line test. What can you conclude?
Answer: A
Why: Passing the vertical test is exactly the condition for being a function, so it is one. Failing the horizontal test means two inputs share an output, which a function is allowed to do, so it is not one-to-one. The parabola is the standard example of precisely this situation.
Real world
Databases are built on this lesson, and they use its vocabulary almost unchanged.
Discussion prompt
A school database stores each student's identification number and their year group. Explain why the number is used as the key rather than the name, in the language of this lesson.
Hint: Which column is the input, and what would go wrong if two rows shared it?
Answer:
Looking a student up by identification number is a function: each number belongs to exactly one student, so the lookup always has exactly one answer.
Looking up by name is not a function, because two students may share a name and the query would return two rows with no way to choose. Database designers call the first situation a primary key, and the requirement they impose on it — uniqueness — is the definition of a function stated in another vocabulary.
The one-to-one idea appears too. Mapping students to year groups is a function but far from one-to-one, since hundreds share a year. That is exactly why you can go from student to year and not back, which is what the horizontal line test predicts.
Commit first
State your confidence along with your answer.
Predict first
Is the relation pairing each whole number with its remainder on division by 3 a function, and is it one-to-one?
Correct: A function, but not one-to-one.
Why: Every whole number has exactly one remainder on division by 3, so it is a function. But 1, 4 and 7 all leave remainder 1, so many inputs share an output and it is not one-to-one. The last option is impossible for any relation at all, since being one-to-one is an extra condition on top of being a function, not an alternative to it.
Explain it
The test of understanding is being able to say why, not just what.
Discussion prompt
In your own words, explain to a classmate why a function is allowed to send two inputs to the same output but not one input to two outputs. Use an everyday example.
Hint: What would you not be able to write down if one input had two outputs?
Answer:
A good explanation makes the asymmetry feel necessary rather than arbitrary. The birthday example works: everyone has exactly one birthday, so the lookup always works, but many people share a birthday and the reverse lookup gets stuck.
The mathematical reason to insist on it is that f(3) must name a number. If the input 3 had two outputs, that symbol would be ambiguous and could not appear in an equation, so every technique built on the notation would collapse.
A good explanation also says what is not forbidden, because that is where the confusion lives. Sharing an output is ordinary and useful — the constant rule does it as loudly as possible, and it is still a function.
Exit ticket
One honest answer, so the next lesson can start in the right place.
Predict first
Which idea from this lesson would you most want to see again?
Correct: Any of these is a legitimate answer; the useful one is the honest one.
Why: There is no correct choice here. The second and third are the ones that cost the most marks on assessments, and the first is the one that quietly causes the other two, so an answer of the first is often the most accurate self-diagnosis even when the symptoms showed up elsewhere.
Connect it up
One page, drawn from memory, is worth more than rereading the section.
Draw it
Draw a diagram with 'relation' as the outer box, 'function' inside it, and 'one-to-one function' inside that. On each boundary, write the condition you must satisfy to move inward. Then, beside the diagram, draw one small graph that lives in each of the three regions, and write next to each which line test it passes.
If the nested boxes came out in the right order and each graph is in the right region, you have the whole lesson. The rest of Chapter 1 works inside this picture.
Recap
Five things, and the fifth one is the entry ticket to Section 1.7.
| if you remember one thing | it should be this |
|---|---|
| about the definition | one input, exactly one output — and nothing about outputs |
| about the notation | f(x) is a name, so nothing distributes over it |
| about the two tasks | which letter is known decides which task you are doing |
| about the two tests | vertical decides function; horizontal decides invertible |
Section 1.2 takes the two sets this lesson named in passing — the domain and the range — and asks how to find them from a rule rather than read them off a list.
OpenStax, Precalculus, §1.1 Functions and Function Notation §1.1, pp. 10-40 — everything on these slides traces back here
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