IB DP Mathematics: Analysis and Approaches -- Functions Strand

This guide covers the Functions strand, which carries 21 hours at SL and 32 at HL for IB Diploma Programme Mathematics: Analysis and Approaches, first assessment 2021. The full syllabus guide positions Functions deliberately before Calculus (28 hours SL / 55 hours HL) in the course structure, since nearly every calculus technique assumes fluent, confident work with functions first.

Where this fits in the syllabus

Differentiation and integration are, in essence, operations performed on functions. A student who is shaky on transforming, graphing or solving equations involving a function will find that weakness resurfaces every time a Calculus question wraps a function in a derivative or integral. Treating Functions purely as a stand-alone topic, rather than as the toolkit the rest of the SL/HL course depends on, is the single most common structural mistake in how students sequence their own revision.

Syllabus coverage

IB DP MATHEMATICS: ANALYSIS AND APPROACHES – FUNCTIONS

• The concept of a function – domain, range, and function notation $f(x)$, including composite functions $f \circ g(x)$ and the conditions needed for an inverse function $f^{-1}(x)$ to exist
• Graphing and transforming functions – translations, stretches and reflections applied to a base graph, and reading key features (intercepts, turning points, asymptotes) directly from a graph or its equation
• Solving equations involving functions – both algebraically and graphically, using technology to find intersections, since Analysis and Approaches expects fluency in both routes
• Types of function most commonly assessed – linear, quadratic, exponential, logarithmic, and rational functions, each with a recognisable graph shape and characteristic transformations

Transformations at a glance

Transformation	Effect on $y = f(x)$
$y = f(x) + k$	Vertical translation by $k$
$y = f(x - a)$	Horizontal translation by $a$
$y = k f(x)$	Vertical stretch, scale factor $k$
$y = f(kx)$	Horizontal stretch, scale factor $1/k$
$y = -f(x)$	Reflection in the $x$-axis
$y = f(-x)$	Reflection in the $y$-axis

Apply transformations to the key features of the base graph (intercepts, turning points, asymptotes) directly, rather than replotting the whole curve from scratch every time – this is far faster under exam time pressure and less error-prone for multi-step transformations.

Worked example: composite and inverse functions

Given $f(x) = 2x + 3$ and $g(x) = x^2$, find $f \circ g(x)$ and explain why $f^{-1}(x)$ exists but a general inverse of $g(x)$ over all real $x$ does not. Substituting $g(x)$ into $f$ gives $f(g(x)) = f(x^2) = 2x^2 + 3$. $f$ has an inverse because it is a strictly increasing linear function – one-to-one over its whole domain – so each output corresponds to exactly one input, which is the condition an inverse function requires. $g(x) = x^2$ does not, in general, have an inverse over all real numbers, because it is not one-to-one: both $x = 2$ and $x = -2$ give $g(x) = 4$, so no single inverse function can recover the original input without restricting the domain first. This one-to-one reasoning – not just mechanically “swapping x and y” – is what Analysis and Approaches mark schemes reward when a question asks why an inverse does or does not exist.

Logarithmic and exponential pairs

Exponential and logarithmic functions are inverses of each other, and this strand expects fluent movement between the two forms: $y = a^x$ rewritten as $x = \log_a y$, and the graph of a logarithmic function recognised as the reflection of the corresponding exponential graph in the line $y = x$ – the general graphical signature of any function and its inverse. Questions combining exponential growth/decay models with logarithms to solve for an unknown exponent are common, and rely on this inverse relationship rather than a separate memorised technique.

How to approach it

Confirm a function is one-to-one before finding its inverse, rather than swapping variables by default – examiners specifically reward this justification, not just the mechanical swap. Practise sketching each transformation from the table above onto a simple base graph such as $y = x^2$, applying it to key features rather than replotting from scratch. Convert confidently between exponential and logarithmic forms of the same relationship, since this fluency underpins a recurring question type. Cross-check an algebraic solution to an equation involving functions against a graphical solution using technology, since Paper 2 allows this directly and a mismatch between the two is a useful signal that an error has been made somewhere in the working. Because Functions underpins Calculus so directly, treat weak function fluency as a Calculus-readiness problem, not just a Functions-topic problem, and revise the two strands together rather than in strict isolation.

Exam traps

Swapping $x$ and $y$ mechanically to find an inverse without first checking the function is one-to-one over the domain given. Applying a horizontal transformation ($f(x-a)$ or $f(kx)$) in the wrong direction relative to the sign given – $f(x - a)$ shifts right, not left, a common source of sign errors. Reading off asymptotes from a transformed graph without checking how the transformation moved them. Treating a graphical solution (technology) and an algebraic solution as interchangeable without checking both give a consistent answer, since Paper 2 explicitly allows technology for exactly this cross-check.

Official syllabus

International Baccalaureate Organization, Diploma Programme Subject Brief – Mathematics: Analysis and Approaches, first assessment 2021 – the same source already cited by the full syllabus guide and the revision notes already on the site. Verified 2026-09-06.