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06062025–2027 syllabus

ADDITIONAL MATHEMATICS · CHAPTER 1

Functions

Treat a function as a complete mapping: its rule, permitted inputs, possible outputs and order of operations all matter.

Additional Mathematics4 connected sectionsSyllabus-aligned guide

LEARNING OBJECTIVES

What you will be able to do

  • identify functions, one-to-one and many-to-one mappings
  • find domains and ranges with algebraic restrictions
  • construct inverse and composite functions
  • connect modulus and inverse functions to their graphs

AT A GLANCE

Syllabus0606Coverage2025–2027Sections4LevelAdditional Mathematics

INTRODUCTION · THE BIG IDEA

Treat a function as a complete mapping: its rule, permitted inputs, possible outputs and order of operations all matter.

Functions provide the language used throughout Additional Mathematics. A formula alone is not the whole function: the domain can decide whether an inverse or a composite is valid.

Questions often combine notation, restrictions and graphs. State each restriction before simplifying so that an algebraically neat answer does not hide an invalid input.

01

SECTION 01

Mappings, domain and range

Core concept

The domain is the allowed input set and the range is the output set actually produced. A many-to-one rule may still be a function, but it cannot have an inverse over that domain.

DETAILED EXPLANATION

  • A square-root expression must be non-negative for real outputs.
  • A logarithm argument must be strictly positive.
  • Use interval or inequality notation consistently.
Common real-domain restrictions
FeatureRestrictionReason
1/g(x)g(x) ≠ 0division by zero is undefined
√g(x)g(x) ≥ 0real square roots need non-negative inputs
ln g(x)g(x) > 0real logarithms need positive arguments
02

SECTION 02

Inverse functions

Core concept

An inverse reverses the original mapping. To find it, write y = f(x), rearrange for x, then interchange x and y. Restrict the original domain first if the rule is not one-to-one.

RULE 1
f⁻¹(f(x)) = x for x in the domain of f
RULE 2
f(f⁻¹(x)) = x for x in the range of f
Typical linear, quadratic, cubic, reciprocal and exponential graph shapeslinearquadraticcubicreciprocalexponential
Original graph-family guide: use intercepts, symmetry, turning points and asymptotes to identify each family.
Original worked example

Reverse an exponential mapping

  1. Let f(x) = e^(2x−1). Write y = e^(2x−1).
  2. Take natural logarithms: ln y = 2x − 1.
  3. Rearrange: x = (ln y + 1)/2.
  4. Exchange the variables and state the inverse domain x > 0.

Answer: f⁻¹(x) = (ln x + 1)/2, for x > 0.

03

SECTION 03

Composite and repeated functions

Core concept

In fg(x), apply g first and then f. The output of the inner function must lie in the domain of the outer function, so composition can introduce new restrictions.

DETAILED EXPLANATION

  • f²(x) means f(f(x)), not [f(x)]².
  • For gf to exist, each selected output of f must be accepted by g.
RULE 1
Domain(gf) ⊆ Domain(f)
RULE 2
Range(gf) ⊆ Range(g)
ORIGINAL STUDY DIAGRAMBuild a composite safely
1Start with the inner input
2Apply the inner rule
3Check the outer domain
4Apply the outer rule
5Simplify with restrictions kept
04

SECTION 04

Modulus and inverse graphs

Core concept

For y = |f(x)|, keep the part of y = f(x) on or above the x-axis and reflect every negative part upward. A function and its inverse have graphs reflected in y = x.

DETAILED EXPLANATION

  • Zeros stay fixed under the modulus transformation.
  • A reflected section can create sharp corners.
  • The domain and range swap between a one-to-one function and its inverse.
  • The graphs of inverse functions meet on y = x where f(x) = x.
RULE 1
|f(x)| = f(x) when f(x) ≥ 0
RULE 2
|f(x)| = −f(x) when f(x) < 0

QUICK CHAPTER SUMMARY

The ideas to carry forward

  • A function gives one output for every allowed input.
  • Domain restrictions are part of the answer.
  • Inverse functions reverse a one-to-one mapping and reflect in y = x.
  • Composite notation is read from right to left.
  • Modulus reflects negative graph sections above the x-axis.

QUICK REVISION CHECKLIST

Can you do each of these without your notes?

  • identify functions, one-to-one and many-to-one mappings
  • find domains and ranges with algebraic restrictions
  • construct inverse and composite functions
  • connect modulus and inverse functions to their graphs
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