Functions

Visual model

A function maps each input to an output

input

f(x)

output

Gold-standard guide

26 mins

What you will learn

Use function notation, mappings and inverse operations.

Use a clear step-by-step method for functions.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier algebra skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

f(x) means 'do this operation to x'

Step 1

Evaluate f(3)

f(3) = 2(3) − 1 = 5

Step 2

Evaluate f(−2)

f(−2) = 2(−2) − 1 = −5

Step 3

Find the inverse: write y = 2x − 1

Swap x and y: x = 2y − 1

Watch out

For composite functions, students apply f first then g

Function notation

f(3) means put 3 into the function f.

Composite function

fg(x) means apply g first, then f.

Worked example

Given f(x) = 2x − 1, find f(3), f(−2) and the inverse function f^-1(x).

Evaluate f(3): f(3) = 2(3) − 1 = 5.

Evaluate f(−2): f(−2) = 2(−2) − 1 = −5.

Find the inverse: write y = 2x − 1: Swap x and y: x = 2y − 1.

Rearrange for y: 2y = x + 1 → y = (x + 1)/2. So f^-1(x) = (x + 1)/2.

Final answer

f(3) = 5, f(−2) = −5, f^-1(x) = (x + 1)/2

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Given f(x) = 2x − 1, find f(3), f(−2) and the inverse function f^-1(x).

4 marks4 minsfunctions-worked

Show solution

Worked solution

1.Evaluate f(3): f(3) = 2(3) − 1 = 5.
2.Evaluate f(−2): f(−2) = 2(−2) − 1 = −5.
3.Find the inverse: write y = 2x − 1: Swap x and y: x = 2y − 1.
4.Rearrange for y: 2y = x + 1 → y = (x + 1)/2. So f^-1(x) = (x + 1)/2.

Final answer

f(3) = 5, f(−2) = −5, f^-1(x) = (x + 1)/2

Mark points

M1: evaluate f(3)
M1: evaluate f(−2)
M1: find the inverse: write y = 2x − 1
M1: rearrange for y
A1: f(3) = 5, f(−2) = −5, f^-1(x) = (x + 1)/2

Watch out

For composite functions, students apply f first then g. Remember fg(x) means 'g then f' — work from right to left.Also, the domain of f^-1 may differ from f's domain; flag any restrictions.

Diagnosticrecall

Given g(x) = x² + 1, find g(−3).

1 mark2 minsfunctions-q1

Show solution

Worked solution

1.Spot the skill: f(x) means 'do this operation to x'.
2.Use the evaluate f(3) stage first, then evaluate f(−2).
3.Keep the final answer visible: 10.

Final answer

Mark points

M1: use the correct f(x) means 'do this operation to x'. substitute directly for f(a).composite fg(x) means apply g first, then f.inverse f^-1 reverses f: write y = f(x), swap x and y, rearrange for the new y.
A1: 10

Watch out

For composite functions, students apply f first then g. Remember fg(x) means 'g then f' — work from right to left.Also, the domain of f^-1 may differ from f's domain; flag any restrictions.

Easyprocedure

Find fg(2) where f(x) = 3x and g(x) = x + 4.

2 marks3 minsfunctions-q2

Show solution

Worked solution

1.Spot the skill: f(x) means 'do this operation to x'.
2.Use the evaluate f(−2) stage first, then find the inverse: write y = 2x − 1.
3.Keep the final answer visible: f(g(2)) = f(6) = 18.

Final answer

f(g(2)) = f(6) = 18

Mark points

M1: use the correct f(x) means 'do this operation to x'. substitute directly for f(a).composite fg(x) means apply g first, then f.inverse f^-1 reverses f: write y = f(x), swap x and y, rearrange for the new y.
A1: f(g(2)) = f(6) = 18

Watch out

For composite functions, students apply f first then g. Remember fg(x) means 'g then f' — work from right to left.Also, the domain of f^-1 may differ from f's domain; flag any restrictions.

Mediumreasoning

Find the inverse of h(x) = (x − 3)/5.

3 marks4 minsfunctions-q3

Show solution

Worked solution

1.Spot the skill: f(x) means 'do this operation to x'.
2.Use the find the inverse: write y = 2x − 1 stage first, then rearrange for y.
3.Keep the final answer visible: h^-1(x) = 5x + 3.

Final answer

h^-1(x) = 5x + 3

Mark points

M1: use the correct f(x) means 'do this operation to x'. substitute directly for f(a).composite fg(x) means apply g first, then f.inverse f^-1 reverses f: write y = f(x), swap x and y, rearrange for the new y.
A1: h^-1(x) = 5x + 3

Watch out

For composite functions, students apply f first then g. Remember fg(x) means 'g then f' — work from right to left.Also, the domain of f^-1 may differ from f's domain; flag any restrictions.

Hardproblem solving

Solve f(x) = 11 where f(x) = 4x − 1.

3 marks5 minsfunctions-q4

Show solution

Worked solution

1.Spot the skill: f(x) means 'do this operation to x'.
2.Use the rearrange for y stage first, then evaluate f(3).
3.Keep the final answer visible: x = 3.

Final answer

x = 3

Mark points

M1: use the correct f(x) means 'do this operation to x'. substitute directly for f(a).composite fg(x) means apply g first, then f.inverse f^-1 reverses f: write y = f(x), swap x and y, rearrange for the new y.
A1: x = 3

Watch out

For composite functions, students apply f first then g. Remember fg(x) means 'g then f' — work from right to left.Also, the domain of f^-1 may differ from f's domain; flag any restrictions.

Exam-stylemulti-step

Given f(x) = x² − 1 and g(x) = 2x, find gf(3).

4 marks6 minsfunctions-q5

Show solution

Worked solution

1.Spot the skill: f(x) means 'do this operation to x'.
2.Use the evaluate f(3) stage first, then evaluate f(−2).
3.Keep the final answer visible: g(f(3)) = g(8) = 16.

Final answer

g(f(3)) = g(8) = 16

Mark points

M1: use the correct f(x) means 'do this operation to x'. substitute directly for f(a).composite fg(x) means apply g first, then f.inverse f^-1 reverses f: write y = f(x), swap x and y, rearrange for the new y.
A1: g(f(3)) = g(8) = 16

Watch out

For composite functions, students apply f first then g. Remember fg(x) means 'g then f' — work from right to left.Also, the domain of f^-1 may differ from f's domain; flag any restrictions.

Grade 9 stretchproblem solving

Given f(x) = (2x - 3)/(x + 1), x != -1, find f^-1(x) and solve f^-1(x) = 5.

4 marks7 minsfunctions-g9

Show solution

Worked solution

1.Write y = (2x - 3)/(x + 1) and make x the subject.
2.Swap x and y to state the inverse function.
3.Solve the inverse equation and check the value is in its domain.

Final answer

f^-1(x) = (x + 3)/(2 - x), x != 2; solving f^-1(x) = 5 gives x = $\frac{7}{3}$

Mark points

M1: obtain yx + y = 2x - 3
M1: rearrange to x = (y + 3)/(2 - y)
A1: state the inverse with x != 2
A1: solve (x + 3)/(2 - x) = 5 to get x = $\frac{7}{3}$

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

16 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Functions - 2 marksGiven g(x) = x² + 1, find g(−3).Mark answer

Answer

2Algebraic notation and substitution - 2 marksFind t² + 4t when t = −2Mark answer

Answer

−4

3Simplifying expressions - 2 marksSimplify 2xy + 3x²y − xy + x²yMark answer

Answer

3x²y + xy

4Expanding and factorising - 3 marksFactorise fully 6x² + 9xMark answer

Answer

3x(2x + 3)

Mastery check

I can explain the method for functions.
I can show clear working without skipping key steps.
I can avoid this mistake: For composite functions, students apply f first then g.Remember fg(x) means 'g then f' — work from right to left.Also, the domain of f^-1 may differ from f's domain; flag any restrictions.

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Explain functions from the beginning.Give me a hint for a functions question.Quiz me on functions.

Get help with anything that still feels tricky.

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A function maps each input to an output

What you will learn

Know the rule, then use it

Method

Evaluate f(3)

Evaluate f(−2)

Find the inverse: write y = 2x − 1

Watch out

Given f(x) = 2x − 1, find f(3), f(−2) and the inverse function f-1(x).

Build up to the hardest questions

Switch between skills

Get help with anything that still feels tricky.

Given f(x) = 2x − 1, find f(3), f(−2) and the inverse function f^-1(x).