1.02PureFoundation

Function Notation

Q: Is f(x) the same as y?

For graphing, yes — the graph of f is y = f(x). The notation is more flexible because it names the rule and lets you write inputs other than x.

Q: How do I show a function is many-to-one?

Find two different inputs with the same output, such as f(2) = f(-2) for f(x) = x^2.

Q: Does the letter matter?

No. f, g and h are conventional names, and f(t) = t^2 defines the same function as f(x) = x^2.

You have been writing $y = x^{2} + 3$ for years. Function notation replaces it with $f (x) = x^{2} + 3$ , and the change is more than cosmetic: it names the rule, so you can say $f (5)$ , $f (a + h)$ or $f (2 x)$ without rewriting anything, and it forces you to think about which inputs the rule is actually allowed to take.

30 min Functions

The big picture

A $function$ is a rule that assigns to every input in a set exactly $one$ output. That single word "one" is what separates a function from a general relationship between $x$ and $y$ , and it is why a circle is not the graph of a function while a parabola is. Notation matters because the rest of this chapter — domain and range, composition, inverses — and much of the course after it, from graph transformations to differentiation from first principles, is written in it. Being fluent with expressions like $f (x + h) - f (x)$ now saves real confusion later.

What you'll be able to do

Use function notation $f (x)$ and the mapping notation $f : x \mapsto \dots$
Evaluate functions for numbers and for algebraic inputs
Distinguish functions from mappings that are not functions
Describe one-to-one and many-to-one functions

What function notation says

$f (x) = 3 x - 1$ reads "f of $x$ equals $3 x - 1$ ". The letter $f$ names the rule; the $x$ in brackets is a placeholder for whatever you feed in.

The same rule can also be written in mapping form, $f : x \mapsto 3 x - 1$ , read "f maps $x$ to $3 x - 1$ ". OCR uses both, and they mean exactly the same thing.

To evaluate, replace $every$ $x$ with the input — in brackets, so that signs and powers behave. $f (- 2) = 3 (- 2) - 1 = - 7$ .

f (x) = 3 x - 1 ⟺ f : x \mapsto 3 x - 1

Two notations, one rule.

g (- 3) = (- 3)^{2} - 4 (- 3) = 9 + 12 = 21

g (x) = 5

means

x^{2} - 4 x = 5

, so

x^{2} - 4 x - 5 = 0

3Factorise:

(x - 5) (x + 1) = 0

Answer

g (- 3) = 21

;

x = 5

x = - 1

Why this works

$g (- 3)$ asks "what comes out when $- 3$ goes in?" and $g (x) = 5$ asks "what went in if $5$ came out?". The first is evaluation; the second is an equation. Keeping those two questions distinct is half of this topic.

Algebraic inputs

The input does not have to be a number. $f (2 x)$ , $f (x + 1)$ and $f (a + h)$ are all legitimate, and they are evaluated exactly the same way: substitute the whole expression wherever $x$ appears.

This is where brackets earn their keep. With $f (x) = x^{2}$ , $f (x + 1) = (x + 1)^{2} = x^{2} + 2 x + 1$ — not $x^{2} + 1$ .

Notice also that $f (x) + 1$ and $f (x + 1)$ are different: one adds after applying the rule, the other adds before. That distinction becomes vertical versus horizontal translation when you study graph transformations.

f (x + h) = 2 (x + h)^{2} - (x + h) = 2 x^{2} + 4 x h + 2 h^{2} - x - h

2Subtract

f (x) = 2 x^{2} - x

3The

2 x^{2}

and

- x

terms cancel.

Answer

4 x h + 2 h^{2} - h

Tip — Always put the substituted expression in brackets on the first line of working. Most errors with $f (x + h)$ or $f (- x)$ come from skipping this.

When is a mapping a function?

A $mapping$ is any rule linking inputs to outputs. It is a $function$ only if every input in the chosen set produces $exactly one$ output.

So $x \mapsto \pm x$ is not a function: the input $4$ gives two outputs, $2$ and $- 2$ . But $x \mapsto x$ , which by convention means the non-negative root, is a function on $x \geq 0$ .

Graphically, this is the $vertical line test$ : a graph represents a function if no vertical line crosses it more than once. A circle fails; a parabola $y = x^{2}$ passes.

A mapping can also fail to be a function because some input has $no$ output. $x \mapsto \frac{1}{x}$ is not a function on all real numbers, because $0$ has nowhere to go — but it is a function once $0$ is excluded.

Why this works

Whether something is a function depends on the set of inputs you allow, not only on the formula. That is exactly why the next lesson, on domain and range, exists.

One-to-one and many-to-one

A function is $one-to-one$ if different inputs always give different outputs. $f (x) = 2 x + 3$ is one-to-one: no two values of $x$ give the same answer.

It is $many-to-one$ if at least two inputs share an output. $f (x) = x^{2}$ on all real numbers is many-to-one, since $f (3) = f (- 3) = 9$ .

Graphically, a one-to-one function passes the $horizontal line test$ too: no horizontal line meets its graph more than once.

This classification matters because only one-to-one functions have inverses. Restricting a many-to-one function to part of its domain — $x^{2}$ on $x \geq 0$ , say — can make it one-to-one.

Tip — Mappings can be one-to-many (not functions at all), one-to-one, or many-to-one. Only the last two are functions. Examiners like asking you to classify all four combinations.

Think like an examiner

Substitute into $f$ with brackets, especially for negative or algebraic inputs.
When asked whether a mapping is a function, name a specific input that fails and say why.
Distinguish $f (x + a)$ from $f (x) + a$ explicitly.
Use the precise words one-to-one and many-to-one, justified with an example of two inputs.

Common misconceptions

Reading

f (x)

f

multiplied by

x

f (x)

means the rule

f

applied to

x

Writing

f (x + 1) = x^{2} + 1

when

f (x) = x^{2}

Substitute the whole input:

(x + 1)^{2}

Evaluating

f (- 2)

for

f (x) = x^{2}

- 4

(- 2)^{2} = 4

; brackets keep the sign inside the square.

Calling

x \mapsto \pm x

a function.

One input gives two outputs, so it is a one-to-many mapping.

Function notation

f : x \mapsto f (x)

Mapping notation.

f (a) — output for input a

Evaluation.

f (x + a) \neq = f (x) + a

Inside versus outside.

Stretch yourself

The function $f$ is defined by $f (x) = a x + b$ for constants $a$ and $b$ . Given that $f (2) = 11$ and $f (f (1)) = 29$ , find the possible pairs $(a, b)$ .

Hint — Write $f (1) = a + b$ , then apply $f$ to that expression. Use $f (2) = 11$ to eliminate $b$ .

Questions students ask

f (x)

the same as

y

For graphing, yes — the graph of

f

y = f (x)

. The notation is more flexible because it names the rule and lets you write inputs other than

x

What is the difference between a mapping and a function?

Every function is a mapping, but a mapping is only a function if each input in its domain has exactly one output.

How do I show a function is many-to-one?

Find two different inputs with the same output, such as

f (2) = f (- 2)

for

f (x) = x^{2}

Does the letter matter?

No.

f

g

and

h

are conventional names, and

f (t) = t^{2}

defines the same function as

f (x) = x^{2}

Key takeaways

$f (x)$ names a rule and its input; $f : x \mapsto \dots$ is equivalent.
Substitute the whole input, in brackets, wherever $x$ appears.
A function gives exactly one output for each input in its domain.
One-to-one: distinct inputs, distinct outputs. Many-to-one: some outputs are shared.
Only one-to-one functions have inverses.

How this fits the course

Build on

Factorising Expressions

Parametric Equations

Leads to

Domain and Range

Test yourself

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Real past-paper questions on Function Notation, marked mark-by-mark. How you do feeds straight into your weak-topic list, so your revision keeps targeting what actually needs work.

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Function Notation

What function notation says

Algebraic inputs

When is a mapping a function?

One-to-one and many-to-one

Practise this topic

Related topics

Function Notation

What function notation says

Algebraic inputs

When is a mapping a function?

One-to-one and many-to-one

Practise this topic

Related topics