J560 · 6.05–6.06Sequences & FunctionsHigherGrades 7–9

Functions

A function is a rule that takes an input and produces exactly one output — like a machine. Writing $\mathrm{f}(x) = 3x - 2$ gives the machine a name, so you can ask for $\mathrm{f}(4)$ , 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. $\mathrm{f}(4)$ means "put 4 into the rule f"; $\mathrm{fg}(x)$ 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 $\mathrm{f}^{-1}$ undoes f. Finding it is rearranging a formula, and OCR often combines all three ideas in one question — evaluate, compose, then solve an equation involving an inverse.

Inverse Functions - CorbettmathsCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

Before you start

Try a short calculation from the preceding topic without the notes.
Check the tier and grade band shown in this lesson.

Watch, pause, try

Video by corbettmaths. The video’s own board label is retained; follow the specification reference and scope notes on this lesson for assessed content.

Viewing focus: Use function notation to evaluate functions and solve equations

After watching, attempt these questions before opening the answers. You can also use the written lesson below.

After watching 1: $\mathrm{f}(x) = 5 - 2x$ . What is $\mathrm{f}(-3)$ ?

Show answer and reasoning

$11$ . $5 - 2(-3) = 11$ .

After watching 2: $\mathrm{g}(x) = x^{2} - 4$ . Solve $\mathrm{g}(x) = 21$ .

Show answer and reasoning

$x = \pm 5$ . $x^{2} = 25$ .

What you'll learn

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

Function notation

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

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

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

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

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

Your answer

Tip — Read $\mathrm{f}(a) = b$ 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

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

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

In general $\mathrm{fg}(x) \neq \mathrm{gf}(x)$ .

\mathrm{fg}(x) = \mathrm{f}(\mathrm{g}(x))

g first, then f.

Example 1

\mathrm{f}(x) = 3x - 2

and

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

. Find (a)

\mathrm{fg}(2)

(b)

\mathrm{gf}(2)

(c)

\mathrm{fg}(x)

1(a)

\mathrm{g}(2) = 5

, then

\mathrm{f}(5) = 13

2(b)

\mathrm{f}(2) = 4

, then

\mathrm{g}(4) = 17

3(c)

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

Answer(a)

13

(b)

17

(c)

3x^{2} + 1

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

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

Your answer

Why this works

$\mathrm{fg}(2) = 13$ but $\mathrm{gf}(2) = 17$ : 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 $\mathrm{f}^{-1}$ reverses f: if $\mathrm{f}(5) = 13$ then $\mathrm{f}^{-1}(13) = 5$ .

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

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

Example 1Find

\mathrm{f}^{-1}(x)

when

\mathrm{f}(x) = \dfrac{2x + 5}{3}

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

, so

3y = 2x + 5

2x = 3y - 5

, so

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

3Replace

y

with

x

Answer

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

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

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

Your answer

Tip — Check an inverse by composing: $\mathrm{f}^{-1}(\mathrm{f}(1)) = \mathrm{f}^{-1}\left(\tfrac73\right) = \dfrac{7 - 5}{2} = 1$ ✓.

Solving with functions

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

Example 1With

\mathrm{f}(x) = 3x - 2

and

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

, solve

\mathrm{fg}(x) = 28

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

3x^{2} + 1 = 28

, so

x^{2} = 9

Answer

x = 3

x = -3

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

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

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 $\mathrm{f}^{-1}(x)$ .

Functions

\mathrm{fg}(x) = \mathrm{f}(\mathrm{g}(x))

Composite: g first.

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

Inverse undoes f.

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

Finding an inverse.

Watch out for these

Applying f first in

\mathrm{fg}(x)

The function next to

x

, g, acts first.

Writing

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

The inverse reverses the operations:

\dfrac{x + 2}{3}

Substituting without brackets:

3x^{2} + 1 - 2

for

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

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

Giving only

x = 3

for

x^{2} = 9

Include

x = -3

Stretch yourself

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

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

Questions students ask

What does

\mathrm{ff}(x)

mean?

Apply f twice:

\mathrm{f}(\mathrm{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

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

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 $g(x) = 2x + 1$ . Find $fg(x)$ in its simplest form.[2]

Question 2Grade 83 marks

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

Question 3Grade 93 marks

$f(x) = \dfrac{2x + 3}{x - 1}$ . 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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OCR J560 official specification (opens in a new tab)
Check the assessed scope and practical or mathematical requirements.
Read or explore this topic (opens in a new tab)
An external explanation or interactive activity to use alongside the lesson. It may include extension material.

Return to this tomorrow

Close the notes and answer from memory. Revisit any explanation you cannot yet reconstruct.

$\mathrm{f}(x) = x + 3$ , $\mathrm{g}(x) = 2x$ . What is $\mathrm{fg}(4)$ ?
With the same functions, what is $\mathrm{gf}(x)$ ?
$\mathrm{f}(x) = \dfrac{x - 1}{4}$ . What is $\mathrm{f}^{-1}(x)$ ?
$\mathrm{f}(x) = 3x$ and $\mathrm{g}(x) = x - 2$ . Solve $\mathrm{fg}(x) = \mathrm{gf}(x)$ .

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 3 lessons

Functions

A function is a rule that takes an input and produces exactly one output — like a machine. Writing

\mathrm{f}(x) = 3x - 2

gives the machine a name, so you can ask for

\mathrm{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

Watch, pause, try

Video by corbettmaths. The video’s own board label is retained; follow the specification reference and scope notes on this lesson for assessed content.

Viewing focus: Use function notation to evaluate functions and solve equations

After watching, attempt these questions before opening the answers. You can also use the written lesson below.

After watching 1: $\mathrm{f}(x) = 5 - 2x$ . What is $\mathrm{f}(-3)$ ?

Show answer and reasoning

$11$ . $5 - 2(-3) = 11$ .

After watching 2: $\mathrm{g}(x) = x^{2} - 4$ . Solve $\mathrm{g}(x) = 21$ .

Show answer and reasoning

$x = \pm 5$ . $x^{2} = 25$ .

Practise it the way it is examined

Question 1Grade 72 marks

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

Question 2Grade 83 marks

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

Question 3Grade 93 marks

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

Return to this tomorrow

Close the notes and answer from memory. Revisit any explanation you cannot yet reconstruct.

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

\mathrm{g}(x) = 2x

. What is

\mathrm{fg}(4)

With the same functions, what is

\mathrm{gf}(x)

\mathrm{f}(x) = \dfrac{x - 1}{4}

. What is

\mathrm{f}^{-1}(x)

\mathrm{f}(x) = 3x

and

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

. Solve

\mathrm{fg}(x) = \mathrm{gf}(x)