IB Maths: Composite & Inverse Functions FAQ

Most marks are lost on notation order — writing $f(g(x))$ when you mean $g(f(x))$ — and on skipping domain restrictions when finding an inverse. I've marked hundreds of scripts where the algebra is perfect but the final answer ignores that the inverse only exists on a restricted domain, costing an easy A1 mark.

$f^{-1}(x)$ is the inverse function — it undoes what $f$ does, so $f(f^{-1}(x))=x$. $\frac{1}{f(x)}$ is simply one divided by $f(x)$, a completely different reciprocal function. Confusing the two is one of the most penalised notation errors in Paper 1 short-answer questions.

A function only has a true inverse if it's one-to-one (injective) — each output comes from exactly one input. Use the horizontal line test: if any horizontal line crosses the graph more than once, the function isn't one-to-one on that domain and you must restrict the domain first before finding $f^{-1}$.

Write $y=f(x)$, swap $x$ and $y$, then rearrange to make $y$ the subject — that's your $f^{-1}(x)$. Always state the domain of the inverse, which equals the range of the original function. Missing this last step is the single most common way students drop marks on 5-mark inverse questions.

Substitute the entire expression for $g(x)$ into every $x$ in $f(x)$ — work from the inside out. Simplify carefully afterwards, and always check whether the domain of $g$ restricts what values can feed into $f$, since IB Paper 1 questions often hide a domain trap here.

Substitute your answer back in: if $f^{-1}(f(x))=x$ for all valid $x$, you've got it right. On a GDC, graph $f(x)$, $f^{-1}(x)$ and $y=x$ together — a correct inverse always reflects perfectly across the line $y=x$, which is a fast visual check before you commit to a final answer.

Both, but Paper 1 (non-calculator) tests the algebraic manipulation directly — finding $f^{-1}(x)$ or simplifying $f(g(x))$ by hand. Paper 2 tends to embed composites inside larger modelling or graph-analysis questions, so the skill needs to be automatic rather than something you work out from scratch under time pressure.

The core skills — finding composites, finding inverses, checking domain/range — are identical in Analysis and Approaches (AA) and Applications and Interpretation (AI). AA tends to pair it with more algebraic manipulation and exact-value questions; AI leans towards graphical and technology-based interpretation, using your GDC to explore transformations rather than pure algebra.

The basic mechanics are the same at SL and HL, but HL adds extra layers — self-inverse functions, composite functions involving piecewise or rational expressions, and questions that combine inverses with calculus (like differentiating an inverse function). HL students should expect these ideas woven into harder, multi-part questions rather than tested in isolation.

Revise domain and range first — you can't correctly restrict an inverse without understanding them. Then link composite functions to transformations of graphs and, at HL, to differentiation of inverse functions, since IB exams frequently combine these ideas into a single multi-part question worth 6-8 marks.

A-Level Maths (Edexcel/AQA) covers composite and inverse functions with broadly similar depth to IB Maths AA, but IB tends to test the concept more synoptically — combining it with calculus, transformations or modelling in the same question rather than isolating it. Your child won't find the underlying maths harder, but the way it's assessed demands more joined-up thinking.

Struggling with notation alone isn't a strong enough reason to choose AI over AA — both courses require the same composite and inverse function skills. The real deciding factor should be whether your child enjoys pure algebraic proof (AA) or prefers applied, technology-driven problem-solving (AI), since that difference runs through the whole two-year course, not just one topic.

Look for resources with worked, step-by-step examples rather than just definitions — this topic is learned by doing, not reading. Topical worksheets that isolate composite and inverse questions (before mixing them into full past papers) let a student build the algebraic habit before facing exam-style multi-part questions, which is exactly how RevisionPrep structures its Maths worksheets by topic.

This topic is one of the more self-teachable areas of the syllabus — it's rule-based, with clear right and wrong answers, unlike more conceptual topics like proof or statistics inference. A structured topical worksheet with worked solutions, used alongside the official IB Mathematics guide's Topic 2 content, is usually enough for a motivated student to master it independently.