Functions are one of the five main syllabus topics in IB Mathematics: Analysis and Approaches. For exams, the topic centres on a manageable set of ideas: function notation, domain and range, composite and inverse functions, graph features, transformations, equations, and modelling. These ideas also support calculus, so weaknesses in functions often reappear when you differentiate, integrate, or interpret a model.
The most effective preparation combines conceptual understanding with exam-style practice. Learn what each operation does, write enough working to establish your method, and then watch Functions videos and worked solutions to see how those ideas are converted into marks.
What IB Maths AA tests about functions
The official IB subject brief identifies functions as a core syllabus component at both SL and HL. The course places particular emphasis on constructing, communicating, and justifying mathematical arguments, so an answer without a visible method may not demonstrate everything the question assesses.
Functions can appear on any relevant examination paper rather than in a separate functions paper. Under the current assessment model, Paper 1 does not allow technology, while Paper 2 does; HL students also take a technology-required Paper 3 containing extended problem-solving questions. This means you need both algebraic fluency and the ability to use graphing technology intelligently.
The principal function skills are:
evaluating functions using notation such as f(3)
determining domain and range
forming and evaluating composite functions
finding and interpreting an inverse function
identifying intercepts, roots, turning points, and asymptotes
sketching and transforming graphs
solving equations analytically or graphically
working with quadratic, rational, exponential, and logarithmic functions
interpreting functions in mathematical and real-world contexts
A function assigns exactly one output to every permitted input. If f(x) = 2x² - 3, then means substitute 4 for every occurrence of , giving .
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f(4)
x
f(4) = 2(4²) - 3 = 29
Do not confuse f(x) with multiplication. The symbol f names the function, while the expression inside the brackets identifies its input.
Domain and range
The domain is the set of permitted input values. The range is the set of output values actually produced. Unless a question specifies another domain, the IB guide states that the largest possible domain for which the function is defined is normally used.
When finding a real domain, check for these common restrictions:
Expression
Required restriction
Denominator
Cannot equal zero
Even root, such as √u
u ≥ 0
Logarithm, such as ln(u)
u > 0
Stated model
Input may be restricted by context
For example, if h(x) = √(5 - 2x)/(x - 1), then the square root requires x ≤ 5/2, while the denominator requires x ≠ 1. Both conditions must appear in the domain.
Range questions are often more demanding because they require you to think about possible outputs. A graph, turning point, inverse relationship, or algebraic rearrangement can help. For f(x) = (x - 2)² + 3, the minimum output is 3, so the range is f(x) ≥ 3.
Composite and inverse functions
Composite functions
A composite function applies one function and then another. In (f ∘ g)(x) = f(g(x)), apply g first because it is the inner function.
Suppose f(x) = 2x - 3 and g(x) = x² + 1. Then:
(f ∘ g)(x) = f(x² + 1) = 2(x² + 1) - 3 = 2x² - 1
(g ∘ f)(x) = g(2x - 3) = (2x - 3)² + 1
These are different, so composition is generally not commutative. An examiner may also ask for the domain of a composite; every input must be valid for the inner function, and its output must then be accepted by the outer function.
Inverse functions
An inverse function reverses the original mapping. To find one algebraically:
Write y = f(x).
Interchange x and y.
Rearrange to make y the subject.
State the inverse and its domain if relevant.
For f(x) = 3x - 5, write y = 3x - 5, swap variables to obtain x = 3y - 5, and rearrange. Therefore f⁻¹(x) = (x + 5)/3.
An inverse function exists over a given domain only when the original function is one-to-one. For example, f(x) = x² is not one-to-one over all real numbers because 2 and -2 produce the same output. Restricting the domain to x ≥ 0 gives the inverse f⁻¹(x) = √x.
Graphically, a function and its inverse are reflections in the line y = x. Also remember that f⁻¹(x) means inverse, not 1/f(x).
Graphs, transformations, and equations
IB questions frequently ask you to connect an equation with the shape and features of its graph. A complete sketch should show the features requested, which may include axis intercepts, roots, turning points, endpoints, asymptotes, and appropriate behaviour.
Transformations you need to recognise
Starting from y = f(x):
New function
Effect on the graph
f(x) + k
Translate up by k
f(x - h)
Translate right by h
a f(x)
Vertical scale factor `
f(bx)
Horizontal scale factor `1/
-f(x)
Reflect in the x-axis
f(-x)
Reflect in the y-axis
Horizontal transformations cause many errors because the direction or scale appears reversed inside the function. For example, f(x - 4) moves the graph right, not left. The SL transformations practice page provides targeted examples of these question types.
Standard families and their features
For a quadratic, different algebraic forms reveal different information:
For a rational function (ax + b)/(cx + d), where cancellation has not changed the structure, the denominator gives the vertical asymptote x = -d/c, while the ratio of leading coefficients gives the horizontal asymptote y = a/c. A sound sketch also includes intercepts and the position of each branch.
Exponential and logarithmic functions are inverses. The graph of y = aˣ, for a > 0 and a ≠ 1, has domain all real numbers and range y > 0; y = logₐx has domain x > 0 and range all real numbers.
How examiners phrase functions questions
The command term tells you what the final response must contain. Common formulations include:
Findf⁻¹(x) or (f ∘ g)(x) -- show the algebra leading to the result.
State the domain or an asymptote -- a concise answer may be sufficient, but notation must be precise.
Sketch the graph -- label the important features rather than drawing an unscaled generic curve.
Solve an equation -- give all solutions in the stated domain and use the requested accuracy.
Hence find another result -- use the previous part rather than beginning an unrelated method.
Interpret a value -- explain its meaning in the context, including units where appropriate.
On a technology-enabled paper, a question may expect numerical intersections, extrema, or solutions found from a graph. Record the result to the requested degree of accuracy and provide enough evidence, such as the equation solved or the graphs intersected, to make your method clear.
SL and HL distinctions
Both levels study the central language of functions, quadratics, simple rational functions, exponential and logarithmic functions, equations, and transformations. HL extends the topic through additional algebraic and graphical reasoning.
According to the current syllabus structure summarised in the IB Maths AA topic library, HL extensions include factor and remainder theorems, more general rational functions, odd and even functions, self-inverse functions, domain restriction for inverses, inequalities, and modulus graphs. HL students should therefore expect function questions to be combined with proof, polynomial algebra, or unfamiliar problem-solving.
Common mistakes that lose marks
The most frequent errors are procedural rather than conceptual:
applying composite functions in the wrong order
giving an inverse without checking whether the function is one-to-one
treating f⁻¹(x) as a reciprocal
forgetting domain restrictions after algebraic manipulation
reversing horizontal transformations
drawing an asymptote as part of the curve
omitting intercepts or coordinates from a requested sketch
reporting calculator output without showing what was solved
rounding intermediate values too early
Before leaving a question, check whether your answer is mathematically possible. For instance, an inverse's domain should equal the original function's range, and its range should equal the original domain.
An exam-focused revision method
Begin with one subtopic, such as domain and range, rather than attempting all of functions at once. Review the rule, complete several short questions, and then attempt one extended problem in which the skill is combined with graphing or interpretation.
Use the Functions Questionbank to practise by question type. After attempting a problem independently, compare your approach with the available per-question explanation or worked video solution. Watching a method is most useful after you have committed to your own solution because you can identify the exact step where your reasoning diverged.
Keep an error log organised by cause: notation, algebra, domain, graph feature, calculator use, or interpretation. Jojo AI can help explain an incorrect step, but you should then redo the question without assistance. Finish by mixing functions with algebra and calculus questions so that you can recognise the topic when it is not explicitly named.
Conclusion
IB Maths AA functions become manageable when you organise them around a small number of testable ideas: inputs and outputs, composition, inversion, graph features, transformations, and equations. Precise notation and visible working matter because exam questions assess communication and reasoning as well as the final result.
RevisionDojo's Functions Questionbank, Jojo AI feedback, and worked function videos can be combined into a practical cycle: attempt, review the method, correct the error, and attempt a similar question independently.
Emma holds an MMath from the University of Oxford and has taught IB Mathematics for over 20 years, including every year since Analysis & Approaches replaced the old Higher and Standard Level syllabus in 2019. Her focus is IB Mathematics: Analysis & Approaches at SL and HL, developing genuine mathematical intuition from foundational algebra through to the toughest HL topics.
