\((f\circ g)(x)=f(g(x))\)
Work from the inside out: apply \(g\) first, then feed that result into \(f\).
IB MATHEMATICS AA · TOPIC 2 FUNCTIONS · PREMIUM
Composition & Inverse Functions
Functions become much easier when students stop seeing them as isolated formulas and start seeing them as machines. One function takes an input and changes it; composition sends the output through another machine; an inverse tries to undo the journey. This chapter develops that story visually and algebraically before moving into domains, one-to-one restrictions, difficult inverse problems and mixed IB-style applications.
THE FIRST STORY
Composition means one machine feeds another.
Suppose \(g\) changes an input first and then \(f\) acts on the result. The final output is \(f(g(x))\). The order matters: in general, \(f\circ g\neq g\circ f\). The notes begin with diagrams and numerical tables so students can follow the journey before substituting complicated formulas.
Allowed in both stages
An input must first be valid for the inside function and must then produce an output that is allowed by the outside function.
\(f(f(x))\)
Iterating a function creates patterns that can connect naturally to sequences, fixed points and repeated transformations.
Graph after reasoning
Technology is used to inspect intersections and verify behaviour, but the algebraic structure is explained first.
INVERSES UNDO
An inverse reverses the input-output journey.
If \(f(a)=b\), then the inverse reverses that pair: \(f^{-1}(b)=a\). Algebraically, students learn to write \(y=f(x)\), interchange the roles of \(x\) and \(y\), and solve for the new output. The result is then checked by composition rather than accepted blindly.
\[f^{-1}(f(x))=x\qquad\text{and}\qquad f(f^{-1}(x))=x\]
The graph gives the same story geometrically: the graph of \(y=f^{-1}(x)\) is the reflection of \(y=f(x)\) in the line \(y=x\). Coordinates swap, so a point \((a,b)\) becomes \((b,a)\).
WHY DOMAIN RESTRICTIONS MATTER
A quadratic cannot be inverted until we choose one side of the parabola.
For example, \(f(x)=x^2\) on all real numbers is not one-to-one because positive and negative inputs can produce the same output. If we restrict the domain to \(x\ge 0\), each output is produced once and the inverse is \(f^{-1}(x)=\sqrt{x}\). If instead we choose \(x\le 0\), the inverse branch becomes \(f^{-1}(x)=-\sqrt{x}\). The worksheet contains several problems where the correct inverse depends on reading the original domain before doing any algebra.
FULL PARABOLAnot one-to-oneRIGHT BRANCHchoose the positive square-root branchLEFT BRANCHchoose the negative square-root branch
MIXED IB THINKING
The chapter does not stop at routine inverses.
The final examples connect functions with sequences, exponentials, logarithms and the binomial theorem. Students also meet reconstructed IB-style questions involving inverse graphs, restricted domains, intersections, area and tangent conditions. The aim is to make composition and inverse functions useful inside larger problems rather than treat them as a short isolated technique.
INSIDE THE PACK
Explanation, varied practice and full solutions.
25 pages
Narrative explanations, diagrams, tables, graphs, restricted-domain reasoning, worked examples, GDC discussion and difficult mixed synthesis.
21 questions · 11 pages
Composition, domains, inverses, graph reflection, restricted quadratics, GDC, reconstructed IB-style Section B work and a large final challenge.
17 pages
Full algebra, branch selection, domain reasoning, exact values, graphs and detailed solutions to the long linked questions.
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Continue to Exponents & Logarithms
Inverse functions lead naturally into logarithms: the logarithm exists precisely because we need an operation that reverses an exponential function.