HSC Functions: Inverses, Composition, and Graph Sketching Practice

 

Intro

The HSC Functions booklet is a free Mathematics Extension 1 and Extension 2 resource with 45 worked problems covering domains and ranges, inverses, composite functions, transformations, piecewise graphs, and important HSC function families. It is written for Extension 1 students with Extension 2 preview notes; tutors and teachers and designed for structured HSC revision on Vu's Maths Hub.

This deep-dive introduces HSC Functions — browser-readable, aligned with the NESA syllabus.

Summary

The HSC Functions booklet offers a fundamentals review, 45 tiered problems with solutions, appendices, and a conclusion that distils exam habits. Open HSC Functions — no account required. Use this post to plan how to work through a 66-page booklet efficiently.

What is this booklet?

This booklet is written for Extension 1 students with Extension 2 preview notes; tutors and teachers.

Focus: domains and ranges, inverses, composite functions, transformations, piecewise graphs, and important HSC function families.

Topics covered:

• Functions and properties
• Inverses
• Composite functions
• Graphs of inverse functions
• Domain and range
• Even and odd functions
• Transformations
• Piecewise and absolute value

How to use it:

• Read core definitions first; attempt before solutions
• Watch domains and ranges at every composition step
• Look for $y=x$ symmetry when finding inverses
• Use takeaways for pattern recognition

Approximately 66 pages, CC BY 4.0, readable at HSC Functions.

Key fundamental reviews

Core ideas are embedded in the introduction and early problems. Before Part 1, ensure you can handle the following without notes:

• Function notation, domain, and range
• One-to-one functions and inverse existence
• Composition $(f\circ g)(x)$ and order of operations
• Graphs of inverse functions and $y=x$ symmetry
• Even/odd symmetry and transformations
• Piecewise definitions and absolute value graphs

Why fundamentals matter

Function notation, domain, and range — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

One-to-one functions and inverse existence — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Composition $ — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Graphs of inverse functions and $y=x$ symmetry — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Even/odd symmetry and transformations — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Students who skip this section often repeat the same algebra errors in Part 2. Treat HSC Functions fundamentals as a closed-book quiz first.

Problems and how to use them

The HSC Functions booklet packs 45 practice problems into roughly 66 pages — well beyond a single textbook chapter. Each item includes worked solutions; many include Takeaways that highlight the method to reuse in exams.

Overall structure

Part 1 — detailed solutions: basic, medium, and advanced function problems with detailed solutions.

Part 2 — hint-based fluency: warm-up drills, stretch problems, and challenge corner.

Use Part 1 to learn how complete NSW HSC working is written. Use Part 2 in the fortnight before trials — hints are upside-down so you attempt first.

Part 1 (22 problems)

Advanced (5 problems)

This tier contains 5 problems aimed at extension and synthesis. Representative work includes "The Exponential Functional Equation" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.

• The Exponential Functional Equation
• The Harmonic Conjugates of the Golden Segment
• The Four-Function Cycle and Composite Inverses
• Newton's Serpentine and the Hyperbolic Locus
• Inverse Rates for a Quintic Function

Basic (7 problems)

This tier contains 7 problems aimed at foundational fluency. Representative work includes "Rearranging a Relation into a Function" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.

• Rearranging a Relation into a Function
• A Difference Quotient
• A Table and a Quadratic Graph
• Which Quadratic Matches the Graph?
• Four Familiar Functions
• Zeroes and a Sign Table for a Cubic
• A Circle Graph Matching Question

Medium (10 problems)

This tier contains 10 problems aimed at exam-standard reasoning. Representative work includes "The Nested Quadratic Chain" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.

• The Nested Quadratic Chain
• Deriving Functional Rules Through Substitution
• Composites with a Parameter
• A Relation That Splits into Two Functions
• A Family of Lines Through an Intersection Point
• Why a Quadratic Cannot Have Three Distinct Roots
• Symmetry About the Vertex
• A Family of Quadratics and the Locus of the Vertex
• Composition of One-to-One Functions
• Domain Overlaps and One-to-One Testing

Part 2 (23 problems)

Advanced (7 problems)

This tier contains 7 problems aimed at extension and synthesis. Representative work includes "Self-Inverse Rational Functions" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.

• Self-Inverse Rational Functions
• Linear Function Composition
• Advanced Composition Domain
• Investigation of f(x) = |x + 1| + 1x - 1
• The Hyperbolic Divide
• The Intersection of a^x and \log_a x
• Uniform Convergence and the Limit--Integral Trap

Basic (7 problems)

This tier contains 7 problems aimed at foundational fluency. Representative work includes "Evaluation & Domains" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.

• Evaluation & Domains
• Function Tests
• Parity of Functions
• Basic Composition
• Parity of Composite Sine Functions
• The Structural Property of Functional Associativity
• Functional Dilation and Stretching

Medium (9 problems)

This tier contains 9 problems aimed at exam-standard reasoning. Representative work includes "Domain Restriction for Inverse" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.

• Domain Restriction for Inverse
• The Sigmoid Function and Its Inverse
• Piecewise Functions
• Abstract Transformations
• Rational Decomposition and Asymptotic Behavior
• The Geometry of the Modulus Plateau
• Inequalities of Sine Reciprocals
• The Rotated Modulus Region
• Composite and Inverse Domains

Common patterns across the booklet

• Complex numbers: 2 problems — e.g. "The Geometry of the Modulus Plateau"
• Integration: 2 problems — e.g. "Deriving Functional Rules Through Substitution"
• Mechanics: 1 problem — e.g. "The Harmonic Conjugates of the Golden Segment"
• Inequalities: 1 problem — e.g. "Inequalities of Sine Reciprocals"
• Polynomials: 1 problem — e.g. "Why a Quadratic Cannot Have Three Distinct Roots"
• Trigonometry: 1 problem — e.g. "Parity of Composite Sine Functions"

Standout and less-seen problem types

These go beyond routine drills — expect unfamiliar wording or multi-topic synthesis:

• The Exponential Functional Equation: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
• The Harmonic Conjugates of the Golden Segment: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
• The Four-Function Cycle and Composite Inverses: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
• Newton's Serpentine and the Hyperbolic Locus: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
• Inverse Rates for a Quintic Function: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
• Self-Inverse Rational Functions: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.

Working through a large booklet

1. Read core definitions first; attempt before solutions
2. Watch domains and ranges at every composition step
3. Look for $y=x$ symmetry when finding inverses
4. Use takeaways for pattern recognition

Timed practice: allow about one minute per mark; write legible structure even when practising alone. Error log: tag mistakes as concept, algebra, or misreading. Rotation: do not camp on advanced tier if basics still slip.

Open HSC Functions and work steadily — 45 problems is a marathon, not a sprint.

Key appendices

Strict continuity definitions — Formal limit-based continuity (enrichment). Use for quick reference or enrichment beyond routine exam questions.

Open and closed intervals — Domain notation precision. Use for quick reference or enrichment beyond routine exam questions.

Table of variation — Sign-chart method for sketching. Use for quick reference or enrichment beyond routine exam questions.

For HSC preparation, prioritise the first one or two appendices; later entries reward curious students but are not required for standard papers.

Key conclusion

Functions unify Extension 1 and 2. Mastering domain, range, composition, and inverse graphs lets you read and write mathematics fluently across the course.

The booklet's closing section reinforces these habits:

• Read core definitions first; attempt before solutions
• Watch domains and ranges at every composition step
• Look for $y=x$ symmetry when finding inverses
• Use takeaways for pattern recognition

Revisit HSC Functions in the fortnight before trials and redo problems you missed on first pass.

How to study with this booklet

One week per tier in Part 1; use Functions as maintenance while studying calculus topics

General principles:

• Closed-book first: attempt without notes, then check fundamentals.
• Error log: record concept vs algebra vs reading errors.
• Spaced repetition: redo missed questions after 3 and 7 days.
• Past papers last: fix weak topics here, then sit full papers timed.

Mini-FAQ

Who is the HSC Functions booklet for?

Extension 1 students with Extension 2 preview notes; tutors and teachers studying Mathematics Extension 1 and Extension 2 under the NSW HSC.

Should I read solutions before attempting problems?

Attempt Part 1 first. Use Part 2 hints only after a genuine try or partial working.

Where can I read the booklet online?

Open HSC Functions on Vu's Maths Hub — free, no account required.

How many problems are in the booklet?

Roughly 45 practice problems across 66 pages, each with worked solutions.

Is this aligned with NESA?

Topics match Mathematics Extension 1 and Extension 2 outcomes for domains and ranges, inverses, composite functions, transformations, piecewise graphs, and important HSC function families. Confirm scope with your teacher and current NESA documentation.

Common mistakes to avoid

• Finding an inverse without restricting domain for non-one-to-one functions
• Reversing composition order — $(f\circ g)(x)=f(g(x))$, not $f(x)g(x)$
• Graphing $f^{-1}$ by reflecting in the wrong axis
• Ignoring domain restrictions from denominators or square roots
• Rushing to advanced tiers before basic fluency — build foundations first.

Practice on Vu's Maths Hub

Open the free HSC Functions on Vu's Maths Hub — 45 problems with full worked solutions.

Related resources:

• How to use Vu's Maths Hub — Functions booklet chapter order
• HSC Polys Ext 1 — Polynomial function family

More on Vu's Maths Hub

• Browse all booklets
• How to use the booklets
• FAQ
• About the author

All booklets are free for personal and school use under the CC BY 4.0 licence.

Related resources:

• HSC Differential Equations — Functions in modelling
• HSC Trigonometry — Trig as functions

Functions and properties — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, functions and properties routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Functions booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Functions and locate items that stress functions and properties; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Inverses — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, inverses routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Functions booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Functions and locate items that stress inverses; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Composite functions — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, composite functions routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Functions booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Functions and locate items that stress composite functions; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Graphs of inverse functions — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, graphs of inverse functions routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Functions booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Functions and locate items that stress graphs of inverse functions; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Domain and range — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, domain and range routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Functions booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Functions and locate items that stress domain and range; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Even and odd functions — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, even and odd functions routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Functions booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Functions and locate items that stress even and odd functions; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Transformations — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, transformations routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Functions booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Functions and locate items that stress transformations; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Piecewise and absolute value — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, piecewise and absolute value routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Functions booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Functions and locate items that stress piecewise and absolute value; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Syllabus alignment

This booklet supports Mathematics Extension 1 and Extension 2 under the NESA syllabus. It supplements school instruction with 66 pages of extra exam-style practice — not a replacement for class teaching.

Additional exam advice

When sitting Mathematics Extension 1 and Extension 2 exams, allocate time proportional to marks. Practise concise justification in HSC Functions — NSW markers reward clear communication. Reread the booklet conclusion the night before for a habit checklist.

Why Vu's Maths Hub

Vu's Maths Hub hosts every HSC booklet in a continuous, mobile-friendly viewer — zoom, search, no download required. Maintained by Vu Hung Nguyen; CC BY 4.0 for personal and school use.

More on Graphs of inverse functions

Return to HSC Functions and filter mentally for graphs of inverse functions. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.

More on Domain and range

Return to HSC Functions and filter mentally for domain and range. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.