Lesson 2 ~40 min Functions · Path +85 XP

What Makes a Relation a Function

A function is a relation with one extra condition attached. The condition is short, it is easy to check in every representation, and it is the reason the notation of the next lesson is allowed to exist at all.

Today's hook: $y = x^2$ is a function. $x = y^2$ is not. The two equations differ only in which letter is squared, and their graphs are the same curve turned on its side. One condition decides between them, and you can check it by sliding a ruler across the page.

0/5QUESTS

Think First

warm-up

Think of a vending machine. You press a button and something comes out. What would it mean for the machine to be broken in the particular way that a non-function is broken? Not that a button does nothing, but something else. Describe it in one sentence.

Record your answer in your workbook.

The Big Idea

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A function is a relation in which each element of the first set is associated with exactly one element of the second. One input, one output, every time.

$$\text{each first entry appears exactly once}$$

The condition looks different in each representation but is always the same condition. In a list of pairs: no first entry repeats. In a mapping diagram: exactly one arrow leaves each element on the left. On a graph: no vertical line meets the curve more than once.

$\text{one input} \to \text{one output}$

Repeats on the left only

A repeated second entry is fine. A repeated first entry is what breaks it.

One line is enough

To fail the vertical line test you need only find a single line meeting the graph twice.

Many-to-one is allowed

Two inputs may share an output. One input may not have two outputs.

What You'll Master

objectives

Know

That a function is a relation in which each first entry has exactly one partner
How the condition looks in a list of pairs, a mapping diagram and a graph
The vertical line test and what a failure of it demonstrates

Understand

Why a repeated second entry is permitted but a repeated first entry is not
Why the vertical line test is the graphical form of the same condition

Can Do

Decide whether a given relation is a function, in any of the four representations
Apply the vertical line test and justify the verdict
Give a counterexample showing that a relation is not a function

Words You Need

vocabulary

FunctionA relation in which each element of the first set has exactly one partner.

InputA first entry of a pair. The value fed in.

OutputThe second entry paired with a given input.

Vertical line testA graph is a function exactly when no vertical line meets it more than once.

One-to-oneA function in which different inputs always give different outputs.

Many-to-oneA function in which two or more inputs share an output. Still a function.

The Condition

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A relation is a function when every element of the first set is associated with exactly one element of the second.

Two words in that sentence are load-bearing. Exactly rules out an input with two different outputs. One rules out an input with none.

So this relation is a function:

$$\{(1,5),\ (2,5),\ (3,8)\}$$

Each of $1$, $2$ and $3$ appears once as a first entry. The fact that $1$ and $2$ share the output $5$ is entirely acceptable.

This relation is not:

$$\{(1,5),\ (1,7),\ (3,8)\}$$

The input $1$ has two different outputs, so "the output when the input is $1$" has no single answer.

The asymmetry is the whole point. Repeats among the second entries are harmless; a repeat among the first entries, with different partners, is fatal.

Why we care

The next lesson introduces the notation $f(3)$, meaning "the output when the input is $3$". That phrase only makes sense if there is exactly one such output. The function condition is precisely what earns the right to write it.

Checking a List and a Mapping Diagram

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In a list of ordered pairs. Read down the first entries. If any value appears twice with different partners, it is not a function. If a value appears twice with the same partner, the pair was simply written twice, and a set contains no duplicates anyway.

$$\{(0,1),\ (2,3),\ (0,4)\} \quad \text{not a function: } 0 \text{ has two partners}$$

$$\{(0,1),\ (2,3),\ (4,1)\} \quad \text{a function: no first entry repeats}$$

In a mapping diagram. Look at the left region and count the arrows leaving each element. A function has exactly one arrow out of every element on the left. Two arrows leaving one element means two outputs, which fails. No arrow leaving an element also fails, since that input has no output at all.

Nothing is required of the arrows arriving on the right. An element of the second set may receive two arrows, one, or none, and the relation is still a function. The condition constrains the left side only.

The Vertical Line Test

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On a graph, an input is a horizontal position and its outputs are the heights of the points sitting above or below it. All the points with a given input $x$ therefore lie on one vertical line.

So the condition translates directly:

$$\text{a graph is a function} \iff \text{no vertical line meets it more than once}$$

That is the vertical line test. To apply it, imagine sliding a vertical ruler across the whole graph and watching how many times it touches the curve.

Two things about how it is used.

To show a graph is a function you must be satisfied that every vertical line meets it at most once, which is a claim about all of them. To show a graph is not a function you need only exhibit one line that meets it twice. Disproving is cheap; proving needs the general argument.

The circle $x^2 + y^2 = 25$ fails immediately: the line $x = 0$ meets it at $(0,5)$ and $(0,-5)$. Naming those two points is a complete answer.

Which Familiar Graphs Pass

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Functions. Every straight line except a vertical one. Every parabola of the form $y = ax^2 + bx + c$. Every polynomial from the previous focus area, since $y$ is computed from $x$ by a single formula. The hyperbola $y = \tfrac{1}{x}$, and the exponential $y = 2^x$.

Not functions. A vertical line such as $x = 3$, since it contains $(3,y)$ for every $y$ at once. Any circle. The sideways parabola $x = y^2$. The relation $y^2 = x + 1$.

A quick structural signal, though not a proof: if the equation can be rearranged into the form $y = (\text{something in } x)$ with no $\pm$ appearing, it is a function, because the right-hand side computes one value for each $x$. The $\pm$ is the tell:

$$x = y^2 \quad \Longrightarrow \quad y = \pm\sqrt{x}$$

and that $\pm$ is exactly two outputs for each positive input.

Note that $y = \sqrt{x}$, with the positive root only, is a function. The square root symbol denotes the non-negative root by convention, which is a deliberate choice made so that this expression defines a function.

Common Pitfalls

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Rejecting a relation as a non-function because two inputs share an output.

Fix: that is many-to-one, and it is allowed. $y = x^2$ sends both $2$ and $-2$ to $4$ and is a perfectly good function.

Using a horizontal line to test whether a graph is a function.

Fix: the test is vertical, because points sharing an input sit above one another. A horizontal line test answers a different question, about whether the function is one-to-one.

Claiming a graph is a function after checking two or three vertical lines.

Fix: passing requires every line to work. Argue from the shape or from the equation, not from a few samples. Failing, by contrast, needs only one line.

Assuming that any equation involving $x$ and $y$ defines a function.

Fix: it defines a relation. Whether it is a function depends on whether each $x$ produces one $y$, which a circle and a sideways parabola both fail.

Watch Me Solve It · 3 examples

Watch Me Solve It · Deciding from a list

+15 XP per step

PROBLEM

State, with a reason, whether each relation is a function: (a) $\{(1,4),(2,4),(3,4)\}$, (b) $\{(1,4),(1,5),(2,6)\}$, (c) $\{(-2,3),(0,3),(2,-3)\}$.

1

(a) Read down the first entries

$1$, $2$, $3$: no repeats. So each input has exactly one output, and it is a function. The shared output $4$ is irrelevant to the test.
2

(b) Look for a repeated first entry

$(1,4) \ \text{and} \ (1,5)$

The input $1$ has two different outputs, so it is not a function. Naming those two pairs is the complete justification.
3

(c) Check the first entries again

$-2$, $0$, $2$: no repeats, so it is a function. That two outputs are $3$ and one is $-3$ has no bearing.
4

State the pattern being used

In every case the test looked only at the first entries. That is the whole method for a list.

Answer(a) yes; (b) no, since $1$ maps to both $4$ and $5$; (c) yes

Watch Me Solve It · Applying the vertical line test

+15 XP per step

PROBLEM

Decide whether each graph is a function, justifying each answer: (a) the circle $x^2 + y^2 = 9$, (b) the parabola $y = x^2 - 4$, (c) the vertical line $x = 2$.

1

(a) Look for one bad line

$x = 0: \quad y^2 = 9 \ \Rightarrow \ y = \pm 3$

The line $x = 0$ meets the circle at $(0,3)$ and $(0,-3)$, two points. One such line is enough, so it is not a function.
2

(b) Argue that every line works

For any value of $x$, the expression $x^2 - 4$ produces exactly one number, because squaring and subtracting each give a single value. So every vertical line meets the parabola exactly once, and it is a function.
3

(c) Consider the extreme case

The graph of $x = 2$ is itself a vertical line, and it contains the point $(2,y)$ for every real $y$. The vertical line $x = 2$ therefore meets it infinitely often, so it is not a function.
4

Note the asymmetry of the arguments

Parts (a) and (c) named specific failures; part (b) needed a general argument about all values of $x$. Failing is a one-example job; passing is not.

Answer(a) no; (b) yes; (c) no

Watch Me Solve It · Reasoning from an equation

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PROBLEM

Determine whether each equation defines $y$ as a function of $x$: (a) $y = 3x - 7$, (b) $y^2 = x$, (c) $y = \sqrt{x}$, (d) $y = \dfrac{1}{x-2}$.

1

(a) Rearrange if needed and look for ambiguity

The right side computes one value for each $x$, with no $\pm$ and no choice to make. It is a function.
2

(b) Solve for y and watch for the plus-or-minus

$y = \pm\sqrt{x}$

For $x = 4$ this gives $y = 2$ and $y = -2$, two outputs from one input. Not a function.
3

(c) Note the convention on the root symbol

The symbol $\sqrt{\ }$ denotes the non-negative root only, so $\sqrt{4} = 2$ and nothing else. One output per input, so it is a function. Compare with (b): the difference is the $\pm$, not the square root.
4

(d) Check whether any input misbehaves

For every $x$ except $2$ the expression gives exactly one value. At $x = 2$ it gives none, since division by zero is undefined. So it is a function on the inputs where it is defined, and Lesson 4 will say that $2$ is simply excluded from the domain rather than that the function is broken.

Answer(a), (c) and (d) are functions; (b) is not

Brain Trainer · 5 problems

Brain Trainer · Function or not

5 problems

Five items on the function condition. Work each one, then reveal the answer.

1 Is $\{(2,7),(3,7),(4,7)\}$ a function?

No first entry repeats; shared outputs are allowed.Yes
2 Is $\{(5,1),(6,2),(5,3)\}$ a function?

The input $5$ has two different outputs.No
3 Does the graph of $y = x^3$ pass the vertical line test?

Each $x$ gives exactly one cube.Yes
4 Does $x^2 + y^2 = 1$ define a function?

The line $x = 0$ meets it at $(0,1)$ and $(0,-1)$.No
5 Which test decides whether a function is one-to-one?

A different line, checking whether outputs repeat.The horizontal line test

Complete in your workbook.

Multiple Choice · 5 questions

MC1

The condition

+10 XP

A relation is a function when:

MC2

Reading a list

+10 XP

Which of these is NOT a function?

MC3

The test

+10 XP

A graph fails to be a function as soon as:

MC4

Many-to-one

+10 XP

The graph of $y = x^2$ meets the horizontal line $y = 9$ twice. This shows that:

MC5

From an equation

+10 XP

Which equation does NOT define $y$ as a function of $x$?

Short Answer · 3 questions

Decide and justify

+15 XP

SHORT ANSWER

For each relation, state whether it is a function and justify your answer.
(a) $\{(0,0),(1,1),(2,4),(3,9)\}$
(b) $\{(4,2),(4,-2),(9,3)\}$
(c) The graph of $y = 2x - 5$
(d) The graph of $x^2 + y^2 = 16$

Write your working in your book.

The test and its limits

+15 XP

SHORT ANSWER

(a) State the vertical line test and explain why it is equivalent to the definition of a function.
(b) Explain why exhibiting one vertical line that meets a graph twice is enough to disprove that it is a function, but checking three lines that each meet it once is not enough to prove that it is.
(c) The graph of $y = x^2$ meets some horizontal lines twice. Explain why this does not contradict its being a function.
(d) Describe a graph that fails the vertical line test at exactly one value of $x$ and passes everywhere else, and say whether it is a function.

Write your working in your book.

(a) The test: a graph represents a function exactly when no vertical line meets it more than once. It is equivalent to the definition because all the points with a given input $x$ lie on the vertical line through that value; so the number of times that line meets the graph is the number of outputs that input has. The definition demands exactly one output per input, which is exactly the demand that each vertical line meets the graph at most once, and at least once for inputs that are allowed.

(b) The definition is a claim about every input. To disprove a claim of the form "for all", a single counterexample suffices, and a line meeting the graph twice is a counterexample: it exhibits one input with two outputs. To prove such a claim, though, no finite number of examples is enough, since infinitely many inputs remain unchecked and one of them could misbehave. Confirming a function therefore needs a general argument, usually from the equation, that each $x$ produces exactly one $y$.

(c) A horizontal line meeting the graph twice means two different inputs share an output: here $3$ and $-3$ both give $9$. The function condition says nothing about outputs being reused; it constrains only how many outputs each input has. So $y = x^2$ is a function that is many-to-one, and the horizontal line test it fails is the test for being one-to-one, which is a different property.

(d) For example, take the graph of $y = x$ together with the single extra point $(0, 5)$. Every vertical line except $x = 0$ meets it once, and the line $x = 0$ meets it at $(0,0)$ and $(0,5)$. It is not a function, because the definition requires exactly one output for every input and this relation fails at $x = 0$. A single failure anywhere is enough; there is no partial credit in the definition.

Marking guidance: 2 marks for (a). 3 marks for (b): one for the counterexample logic, one for why examples cannot prove, one for what a proof needs. 2 marks for (c). 2 marks for (d).

Functions from equations

+15 XP

SHORT ANSWER

(a) Show that $y^2 = 4x$ does not define $y$ as a function of $x$, by naming a specific input with two outputs.
(b) Does $y^2 = 4x$ define $x$ as a function of $y$? Justify your answer.
(c) Explain why $y = \sqrt{4x}$ is a function while $y^2 = 4x$ is not, given that both involve the same square relationship.
(d) Give an equation whose graph is a function and whose graph is also symmetric about the vertical axis, and one whose graph is a function and is symmetric about the origin.

Write your working in your book.

Stretch Challenge · Cutting a relation down to a function

+25 XP

CHALLENGE

(a) The circle $x^2 + y^2 = 25$ is not a function. Describe two different ways of restricting it so that what remains is a function, and give an equation for each.
(b) Explain why every vertical line fails the vertical line test, and why this is the only straight line that does.
(c) A relation has $m$ pairs and every first entry is different. Explain why it is a function, and state how many pairs a function from a set of $m$ elements must have.

Quick Review

recap

The condition

Each input has exactly one output

In a list

No first entry repeats with a different partner

On a graph

No vertical line meets it more than once

Allowed

Many-to-one. Two inputs may share an output

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