Unit Overview

Functions

IB Mathematics Applications and Interpretation HL· 7 min read 📊 20-25% of overall exam

1. Unit at a Glance

We start with the core definition of a function and key properties that apply to all function types, before building up to core algebraic manipulations and graph transformations. We then explore each major family of functions (quadratic, polynomial, exponential, sinusoidal, rational) in detail, before ending with solving equations and applying functions to real-world modelling tasks. This sequence builds from foundational rules to applied skills, matching how questions are structured on IB exams.

Below are all sub-topics you will cover in this unit:

Concept of functions: domain, range, graphs

Learn the formal definition of a function, how to calculate domain and range, and interpret function graphs.

★⏱ 8 min

Composition of functions and inverse functions

Understand how to combine multiple functions and reverse operations to find inverse functions.

★★⏱ 7 min

Transformations of graphs

Learn how shifts, stretches and reflections transform the graph of any function.

★★⏱ 6 min

Quadratic functions: roots, vertices, inequalities

Explore key properties of quadratics, find roots and vertices, and solve quadratic inequalities.

★★⏱ 8 min

Polynomials: roots, factor theorem, remainder theorem

Master core polynomial theorems and use them to find all roots of higher-order polynomials.

★★★⏱ 9 min

Rational functions and asymptotes

Study rational functions and find all types of asymptotes to sketch their graphs.

★★★⏱ 7 min

Exponential and logarithmic functions

Explore the inverse relationship between exponential and logarithmic functions and their properties.

★★⏱ 8 min

Sinusoidal functions and applications

Learn how sinusoidal functions model periodic phenomena and solve real-world application problems.

★★★⏱ 10 min

Solving equations: algebraic and graphical methods

Compare and use algebraic and graphical approaches to solve a wide range of function equations.

★★⏱ 7 min

Odd and even functions, symmetry

Identify symmetry properties of odd and even functions and use them to simplify function analysis.

★★⏱ 5 min

Inverse functions: algebraic solution

Practice advanced algebraic methods for finding inverse functions of common function types.

★★★⏱ 6 min

Modelling with functions

Learn how to select, fit and validate functions to model real-world data and scenarios.

★★★★⏱ 10 min

2. Common Pitfalls

Wrong move:

Forgetting to restrict domain when finding inverse functions

Why:

Most functions are not one-to-one over their full domain, so inverses are only valid for restricted domains

Correct move:

Always confirm the original function is one-to-one over the given domain before calculating an inverse

Wrong move:

Mixing up horizontal and vertical transformation rules

Why:

Horizontal transformations are counter-intuitive compared to vertical transformations, leading to common sign errors

Correct move:

Remember: $f (x - a)$ shifts right by $a$ , while $f (x) + a$ shifts up by $a$

Wrong move:

Ignoring domain restrictions when simplifying rational functions

Why:

Simplifying cancels points of discontinuity that are still excluded from the original function's domain

Correct move:

Always note excluded $x$ -values from the original function before simplifying

3. Quick Reference Cheatsheet

Concept	Key Formula / Rule
Function definition	A function maps each input in the domain to exactly one output in the range
Inverse function property	$f^{- 1} (f (x)) = x$ , $f (f^{- 1} (x)) = x$ , graph reflects over $y = x$
Vertex of quadratic $f (x) = a x^{2} + b x + c$	$x = - \frac{b}{2 a}$ , $y = f (- \frac{b}{2 a})$
Factor Theorem	$(x - a)$ is a factor of polynomial $p (x)$ if and only if $p (a) = 0$
Exponent-log identity	$lo g_{a} (a^{x}) = x$ , $a^{l o g_{a} x} = x$
General sinusoidal function	$f (x) = a sin (b (x - c)) + d$ , amplitude $∣ a ∣$ , period $\frac{2 π}{∣ b ∣}$
Odd/even function rule	Even: $f (- x) = f (x)$ , symmetric over y-axis; Odd: $f (- x) = - f (x)$ , symmetric over origin

What's Next

Start your study of this unit with the first sub-topic covering the core concept of functions, domain and range. Once you complete all sub-topics in this unit, move on to the next unit on sequences and series, which builds directly on the core function concepts you will learn here.