1MA1 · A7AlgebraHigherGrades 7–9

Functions

A function is a rule that takes an input and produces exactly one output — like a machine. Writing $f (x) = 3 x - 2$ gives the machine a name, so you can ask for $f (4)$ , feed one machine’s output into another, or reverse the machine to find which input produced a given output.

45 min Sequences & Functions

The big picture

Function notation is compact and precise. $f (4)$ means "put 4 into the rule f"; $fg (x)$ means "put $x$ into g, then put the result into f". The order matters, and reading it correctly is where most marks are won or lost.

The inverse function $f^{- 1}$ undoes f. Finding it is rearranging a formula, and Edexcel often combines all three ideas in one question — evaluate, compose, then solve an equation involving an inverse.

What you'll learn

Use function notation to evaluate functions and solve equations
Find composite functions such as $fg (x)$ and $gf (x)$
Find inverse functions $f^{- 1} (x)$
Solve problems combining composite and inverse functions

Function notation

$f (x) = 3 x - 2$ means "the function f multiplies by 3 and subtracts 2".

To find $f (4)$ , substitute: $3 (4) - 2 = 10$ . Negative inputs go in brackets: $f (- 1) = 3 (- 1) - 2 = - 5$ .

$f (x) = 13$ is an equation: $3 x - 2 = 13$ , so $x = 5$ .

Your turnSame method, new numbers — try it before moving on.

$f (x) = 5 - 2 x$ . Find $f (- 3)$ .

Your answer

Tip — Read $f (a) = b$ as "input $a$ gives output $b$ ". Deciding whether a question gives you the input or the output tells you whether to substitute or solve.

Composite functions

$fg (x)$ means $f (g (x))$ : apply g first, then f. The function nearest to $x$ acts first.

Numerically, work from the inside out. Algebraically, substitute the whole expression for $g (x)$ wherever $x$ appears in f.

In general $fg (x) \neq = gf (x)$ .

fg (x) = f (g (x))

g first, then f.

1(a)

g (2) = 5

, then

f (5) = 13

2(b)

f (2) = 4

, then

g (4) = 17

3(c)

f (x^{2} + 1) = 3 (x^{2} + 1) - 2

Answer(a)

13

(b)

17

(c)

3 x^{2} + 1

Your turnSame method, new numbers — try it before moving on.

$f (x) = 2 x + 1$ and $g (x) = x^{2}$ . Find $g f (3)$ .

Your answer

Why this works

$fg (2) = 13$ but $gf (2) = 17$ : the same two machines in a different order give different results, just as "add 1 then square" differs from "square then add 1".

Inverse functions

The inverse $f^{- 1}$ reverses f: if $f (5) = 13$ then $f^{- 1} (13) = 5$ .

To find it, write $y = f (x)$ , rearrange to make $x$ the subject, then replace $y$ with $x$ .

$f^{- 1} (x)$ does not mean $\frac{1}{f ( x )}$ .

y = \frac{2 x + 5}{3}

, so

3 y = 2 x + 5

2 x = 3 y - 5

, so

x = \frac{3 y - 5}{2}

3Replace

y

with

x

Answer

f^{- 1} (x) = \frac{3 x - 5}{2}

Your turnSame method, new numbers — try it before moving on.

$f (x) = \frac{x - 4}{5}$ . Find $f^{- 1} (x)$ .

Your answer

Tip — Check an inverse by composing: $f^{- 1} (f (1)) = f^{- 1} (\frac{7}{3}) = \frac{7 - 5}{2} = 1$ ✓.

Solving with functions

Equations such as $fg (x) = 28$ or $f (x) = f^{- 1} (x)$ are solved by writing each side as an expression first, then solving.

fg (x) = 3 x^{2} + 1

3 x^{2} + 1 = 28

, so

x^{2} = 9

Answer

x = 3

x = - 3

Your turnSame method, new numbers — try it before moving on.

With $f (x) = 2 x + 1$ and $g (x) = x^{2}$ , solve $g f (x) = 25$ .

Your answer

Separate with a comma.

Think like an examiner

Show the inside function evaluated before the outside one.
Use brackets when substituting an expression into a function.
Write inverses in terms of $x$ , using the notation $f^{- 1} (x)$ .

Functions

fg (x) = f (g (x))

Composite: g first.

f^{- 1} (f (x)) = x

Inverse undoes f.

y = f (x) \to x = \dots \to f^{- 1} (x)

Finding an inverse.

Watch out for these

Applying f first in

fg (x)

The function next to

x

, g, acts first.

Writing

f^{- 1} (x) = \frac{1}{3 x - 2}

The inverse reverses the operations:

\frac{x + 2}{3}

Substituting without brackets:

3 x^{2} + 1 - 2

for

f (x^{2} + 1)

3 (x^{2} + 1) - 2 = 3 x^{2} + 1

Giving only

x = 3

for

x^{2} = 9

Include

x = - 3

Stretch yourself

$f (x) = 2 x + 5$ . Solve $f (x) = f^{- 1} (x)$ .

Hint — Find $f^{- 1} (x)$ first, then set the two expressions equal.

Questions students ask

What does

ff (x)

mean?

Apply f twice:

f (f (x))

How do I check a composite function?

Pick a number, evaluate step by step, and compare with substituting into your algebraic answer.

Does every function have an inverse?

At GCSE the functions you invert are one-to-one. Functions like

x^{2}

need a restricted domain, which is A-Level content.

Key takeaways

$f (a)$ : substitute $a$ . $f (x) = b$ : solve for $x$ .
$fg (x) = f (g (x))$ : g first, then f.
Usually $fg (x) \neq = gf (x)$ .
Find $f^{- 1}$ by rearranging $y = f (x)$ for $x$ .
Check inverses by composing: $f^{- 1} (f (x)) = x$ .

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 72 marks

$f (x) = x^{2} - 3$ and $g (x) = 2 x + 1$ . Find $f g (x)$ in its simplest form.[2]

Question 2Grade 83 marks

$f (x) = 3 x + 2$ . Solve $f (x) = f^{- 1} (x)$ .[3]

Question 3Grade 93 marks

$f (x) = \frac{2 x + 3}{x - 1}$ . Find $f^{- 1} (x)$ .[3]

Unlimited practice

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A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

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Functions

Function notation

Composite functions

Inverse functions

Solving with functions

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Functions

Function notation

Composite functions

Inverse functions

Solving with functions

Practise it the way it is examined

Skill drill

Practise this topic

Related topics