Intro
Functions are the language of Extension 2 — transformations, inverses, and domain restrictions reappear in calculus, complex mappings, and graph sketching. Weak function skills make integration bounds and loci questions much harder than they need to be. Keywords: HSC functions theory, Ext 2 math foundation, composite functions. NSW HSC Mathematics.
Summary
Master domain and range, composite functions f(g(x)), inverses with swapped domain/range, and even/odd symmetry. Integration substitutions recognise inner functions; complex mappings w = f(z) reuse the same thinking; mechanics uses position as a function of time.
The Functions booklet on vumaths.com sequences topics from transformations toward Extension 2 depth — follow that order rather than random chapters. Composite and inverse questions reappear in trial papers year after year.
Key Points
- Domain and range — critical for inverse trig and rational functions.
- Composite f(g(x)): order matters; find inner domain first.
- Inverse functions: reflect in y = x; domain ↔ range swap.
- Even/odd symmetry simplifies definite integrals.
- Transformations: shift, scale, reflect — predict graph changes without plotting every point.
- Study in order through the HSC Functions booklet.
Worked example
Question. If f(x) = 1/x for x > 0 and g(x) = x + 2, find the domain of f(g(x)).
Solution.
- f(g(x)) = 1/(x + 2).
- Inner function g(x) = x + 2 is defined for all reals, but f requires positive input: x + 2 > 0.
- Hence x > −2.
Answer. Domain: x > −2.
Takeaway. Domain of a composite is where the inner is defined AND the outer accepts the inner's output.
Exam Preparation
Functions are Year 11 foundation paying off in Year 12 Extension 2. Revisit composites and inverses before starting complex numbers or advanced integration. Sketch one unfamiliar function weekly using domain, intercepts, and asymptotes.
Before each calculus test, spend twenty minutes on composite domains and inverse sketches — these quick wins prevent cascading errors in longer questions. Functions are the grammar of Extension 2; calculus and complex numbers are the literature.
- Composite domain drills. Ten problems finding domain of f(g(x)).
- Inverse pairs. Verify f(f⁻¹(x)) = x on stated domains.
- Link to calculus. Identify inner functions in substitution integrals.
Function fluency accelerates every later topic: integration bounds come from domain restrictions, and inverse functions appear in logarithmic and exponential equations. Spend Year 11 consolidating transformations — shifts, reflections, and dilations — so Year 12 graph sketching is mechanical. When studying Extension 2 complex mappings, revisit how domain constraints on z translate to regions in the Argand plane.
Mini-FAQ
Why do inverse trig functions need restricted domains?
Original trig functions are not one-to-one. Restricting domain makes inverses well-defined — a syllabus requirement.
How do functions connect to complex numbers?
Mappings w = f(z) use the same domain/range language; loci are often level sets of real or imaginary parts.
Should I memorise transformation rules?
Yes — horizontal shift opposite sign to inside (x − h); vertical on outside. Saves time in exams.
Common mistakes to avoid
- Finding f(g(x)) as g(f(x)).
- Inverse domain not swapped from original range.
- Using f(x) = 1/x without excluding x = 0.
- Ignoring even/odd symmetry that could halve integration work.
Schedule function revision before starting new Extension 2 chapters — thirty minutes on composites and inverses saves hours on later integration and complex topics. Sketch one unfamiliar function weekly using domain, intercepts, asymptotes, and symmetry before checking with graphing technology. The Functions booklet chapter order mirrors how topics build — follow it sequentially.
Weak function skills surface as lost marks in integration bounds and complex loci — invest early. Even/odd symmetry can halve definite integral working when domains are symmetric about the origin.
Before each calculus test, spend twenty minutes on composite domains — the highest-yield function revision habit. Inverse functions swap domain and range; write both beside every sketch. Extension 2 complex mappings reuse the same domain thinking as real functions. Transformation rules save time: horizontal shifts opposite sign inside (x − h). Even/odd symmetry can halve definite integral working when domains are symmetric.
Practice on Vu's Maths Hub
Need more practice on this topic? Open the free HSC Functions booklet on Vu's Maths Hub — worked examples and exam-style questions, readable in your browser with no account required.
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All booklets are free for personal and school use under the CC BY 4.0 licence.
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