Functions

What a function actually is

A function is a rule that turns one number into another. Put a number in, get exactly one number out — that “exactly one” is what makes it a function rather than just any old relationship.

You will see functions written in a few ways, all meaning the same thing:

A function machine: $x \to \boxed{\times 3} \to \boxed{+4} \to$ output
A mapping: $x \mapsto 3x + 4$
An equation: $y = 3x + 4$

Building the rule from a machine

Read the machine left to right and do to $x$ exactly what it says, in order.

Machine	Rule
$\times 3$ then $+4$	$3x + 4$
$+4$ then $\times 3$	$3(x + 4)$
$-2$ then $\div 5$	$\dfrac{x - 2}{5}$
$\times 2$ then $+1$ then $\div 3$	$\dfrac{2x + 1}{3}$

Notice rows 1 and 2. Same two operations, different order, completely different function. Try $x = 6$ : the first gives 22, the second gives 30.

Going backwards with inverse operations

To find the input when you know the output, run the machine in reverse and swap every operation for its opposite:

$+ \leftrightarrow - \qquad \times \leftrightarrow \div \qquad \text{square} \leftrightarrow \text{square root}$

Two things have to be reversed: the operations themselves, and the order they happen in. Forgetting the second is the usual slip.

Why this matters later

Inverse operations are not just a sequences topic. They are exactly what you do when you solve an equation:

$4x - 5 = 31 \quad\Rightarrow\quad 4x = 36 \quad\Rightarrow\quad x = 9$

That is the same “add 5, then divide by 4” you did on the machine. Solving equations and reversing function machines are the same skill wearing different clothes — which is why getting comfortable here makes the equations topic much easier.

Key rules and formulas

A function is a rule

One input goes in, exactly one output comes out.

x \mapsto 3 x + 4

means "multiply by 3, then add 4".

Order matters

3 x + 4

and

3 (x + 4)

are different functions. Reading a machine left to right tells you which is which.

Inverse operations

$+ \leftrightarrow - \times \leftrightarrow \div x^{2} \leftrightarrow x$

To undo a function, reverse the machine: go right to left, replacing every operation with its inverse.

Checking an inverse

Put a number through the function, then put the answer through your inverse. If you get your original number back, the inverse is right.

Worked examples

Worked example 1

Reading a function machine

A function machine does

\times 4

then

- 5

. Find the output when the input is 7, and write the function as a rule.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Working backwards

For the function

x \mapsto 4 x - 5

, find the input that gives an output of 31.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Finding the inverse function

Find the inverse of

x \mapsto \frac{x + 6}{2}

Try it yourself first, then reveal the solution one step at a time.

Input, $x$	0	3	5	?
Output	Row 1, 0	Row 1, 3	Row 1, 5	21
Input	0	3	5	Row 2, ?

Input,

x

Output

Input

In this lesson you will learn to…

What a function actually is

Building the rule from a machine

Going backwards with inverse operations

Why this matters later

Key rules and formulas

A function is a rule

Order matters

Inverse operations

Checking an inverse

Worked examples

Reading a function machine

Working backwards

Finding the inverse function

Practice: functions