IB DP Mathematics: Analysis and Approaches -- Functions Strand Revision Notes

Functions carries 21 hours at SL and 32 at HL, as the full syllabus guide sets out, and is deliberately positioned before Calculus (28 hours SL / 55 hours HL) in the course structure: nearly every calculus technique assumes fluent, confident work with functions first. These notes work through the strand’s recurring exam skills, alongside the Calculus strand revision notes and the subject overview already on the site.

Why Functions comes before Calculus

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.

Core skills to hold securely

• 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.

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.

f(g(x)):        substitute g(x) into f: f(x^2) = 2x^2 + 3
Why f has
an inverse:     f is a strictly increasing linear function (one-to-
                one over its whole domain), so each output corresponds
                to exactly one input -- the condition an inverse
                function requires
Why g does
not (in
general):       g(x) = x^2 is not one-to-one over all real numbers --
                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.

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.

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.

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.
• 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.

Quick revision checklist

• Confirm a function is one-to-one before finding its inverse, rather than swapping variables by default.
• Practise sketching each transformation from the table above onto a simple base graph such as $y = x^2$.
• Convert confidently between exponential and logarithmic forms of the same relationship.
• Cross-check an algebraic solution to an equation involving functions against a graphical solution using technology, since Paper 2 allows this directly.

Self-test

1. State the condition a function must satisfy for its inverse to exist.
2. What is the effect of $y = f(x - a)$ on the graph of $y = f(x)$?
3. Why does $g(x) = x^2$ not have an inverse over all real $x$?
4. Why is Functions positioned before Calculus in the course structure?
5. Name two types of function most commonly assessed in this strand.

Answers: 1. The function must be one-to-one (each output corresponds to exactly one input) over the domain considered. 2. A horizontal translation by $a$ units. 3. Because it is not one-to-one over all real numbers – two different inputs (e.g. $2$ and $-2$) can give the same output, so no single inverse function can recover the original input without restricting the domain. 4. Because differentiation and integration operate on functions, so fluency with function notation, graphing and equation-solving underpins nearly every Calculus technique. 5. Any two of: linear, quadratic, exponential, logarithmic, rational.