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 $\mathrm{f}(x) = x^{2} + 3$ , and the change is more than cosmetic: it names the rule, so you can say $\mathrm{f}(5)$ , $\mathrm{f}(a+h)$ or $\mathrm{f}(2x)$ 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 $\textbf{function}$ is a rule that assigns to every input in a set exactly $\textit{one}$ 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 $\mathrm{f}(x+h) - \mathrm{f}(x)$ now saves real confusion later.

What you'll be able to do

Use function notation $\mathrm{f}(x)$ and the mapping notation $\mathrm{f}: x \mapsto \dots$
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

$\mathrm{f}(x) = 3x - 1$ reads "f of $x$ equals $3x - 1$ ". The letter $\mathrm{f}$ 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, $\mathrm{f}: x \mapsto 3x - 1$ , read "f maps $x$ to $3x-1$ ". OCR uses both, and they mean exactly the same thing.

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

\mathrm{f}(x) = 3x - 1 \quad\Longleftrightarrow\quad \mathrm{f}: x \mapsto 3x - 1

Two notations, one rule.

Example 1Given

\mathrm{g}(x) = x^{2} - 4x

, find

\mathrm{g}(-3)

and solve

\mathrm{g}(x) = 5

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

\mathrm{g}(x) = 5

means

x^{2} - 4x = 5

, so

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

3Factorise:

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

Answer

\mathrm{g}(-3) = 21

;

x = 5

x = -1

Why this works

$\mathrm{g}(-3)$ asks "what comes out when $-3$ goes in?" and $\mathrm{g}(x) = 5$ 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. $\mathrm{f}(2x)$ , $\mathrm{f}(x+1)$ and $\mathrm{f}(a+h)$ 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 $\mathrm{f}(x) = x^{2}$ , $\mathrm{f}(x+1) = (x+1)^{2} = x^{2} + 2x + 1$ — not $x^{2} + 1$ .

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

Example 1For

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

, find and simplify

\mathrm{f}(x+h) - \mathrm{f}(x)

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

2Subtract

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

3The

2x^{2}

and

-x

terms cancel.

Answer

4xh + 2h^{2} - h

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

When is a mapping a function?

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

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

Graphically, this is the $\textbf{vertical line test}$ : 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 $\textit{no}$ output. $x \mapsto \dfrac{1}{x}$ 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 $\textbf{one-to-one}$ if different inputs always give different outputs. $\mathrm{f}(x) = 2x + 3$ is one-to-one: no two values of $x$ give the same answer.

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

Graphically, a one-to-one function passes the $\textbf{horizontal line test}$ 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 $x \ge 0$ , 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 $\mathrm{f}$ 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 $\mathrm{f}(x+a)$ from $\mathrm{f}(x)+a$ explicitly.
Use the precise words one-to-one and many-to-one, justified with an example of two inputs.

Common misconceptions

Reading

\mathrm{f}(x)

\mathrm{f}

multiplied by

x

\mathrm{f}(x)

means the rule

\mathrm{f}

applied to

x

Writing

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

when

\mathrm{f}(x) = x^{2}

Substitute the whole input:

(x+1)^{2}

Evaluating

\mathrm{f}(-2)

for

\mathrm{f}(x) = x^{2}

-4

(-2)^{2} = 4

; brackets keep the sign inside the square.

Calling

x \mapsto \pm\sqrt{x}

a function.

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

Function notation

\mathrm{f}: x \mapsto \mathrm{f}(x)

Mapping notation.

\mathrm{f}(a) \text{ — output for input } a

Evaluation.

\mathrm{f}(x+a) \neq \mathrm{f}(x) + a

Inside versus outside.

Stretch yourself

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

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

Reveal solution

SolutionFrom

\mathrm{f}(2) = 11

2a + b = 11

, so

b = 11 - 2a

. Then

\mathrm{f}(1) = a + b = 11 - a

, and

\mathrm{f}(\mathrm{f}(1)) = a(11 - a) + b = 11a - a^{2} + 11 - 2a = -a^{2} + 9a + 11

. Setting this equal to

29

a^{2} - 9a + 18 = 0

, so

(a-3)(a-6) = 0

. If

a = 3

b = 5

; if

a = 6

b = -1

. Check:

\mathrm{f}(x) = 3x+5

gives

\mathrm{f}(1) = 8

\mathrm{f}(8) = 29

✓;

\mathrm{f}(x) = 6x - 1

gives

\mathrm{f}(1) = 5

\mathrm{f}(5) = 29

✓. So

(a,b) = (3, 5)

(6, -1)

\blacksquare

Questions students ask

\mathrm{f}(x)

the same as

y

For graphing, yes — the graph of

\mathrm{f}

y = \mathrm{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

\mathrm{f}(2) = \mathrm{f}(-2)

for

\mathrm{f}(x) = x^{2}

Does the letter matter?

No.

\mathrm{f}

\mathrm{g}

and

\mathrm{h}

are conventional names, and

\mathrm{f}(t) = t^{2}

defines the same function as

\mathrm{f}(x) = x^{2}

Key takeaways

$\mathrm{f}(x)$ names a rule and its input; $\mathrm{f}: x \mapsto \dots$ 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.