Unit Overview

Functions

IB Mathematics: Analysis and Approaches HL· 6 min read 📊 22-24% of overall exam

1. Unit at a glance

We progress sequentially from foundational definitions to advanced HL-only content, building your toolkit step by step. First we cover core definitions and basic manipulations, then we explore each major function family, before ending with general properties and problem solving.

Concepts from this unit are tested heavily across both Paper 1 and Paper 2, and are required for nearly all other units in the course, including calculus, differential equations, and modeling. Mastery of this unit is essential for achieving a high overall score.

Below are all sub-topics in this unit, ordered for sequential learning:

Function definitions and notation

Learn what formally defines a function and standard notation for working with functions.

★⏱ 5 min

Domain and range

Calculate the domain and range for any function, including functions with restricted domains.

★★⏱ 5 min

Composite and inverse functions

Combine functions into composites and find the inverse of a function when it exists.

★★⏱ 7 min

Transformations of functions

Apply shifts, stretches, and reflections to any function and its graph.

★★⏱ 6 min

Linear and quadratic functions

Review and explore key properties of linear and quadratic functions.

★⏱ 5 min

Polynomial and rational functions

Analyze roots, asymptotes, and graphs of polynomial and rational functions.

★★★⏱ 8 min

Exponential and logarithmic functions

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

★★★⏱ 7 min

Basic trigonometric functions

Analyze graphs, domains, ranges, and key properties of the six basic trigonometric functions.

★★★⏱ 6 min

Inverse trigonometric functions (HL only)

HL-only content covering definitions, restricted domains, and properties of inverse trigonometric functions.

★★★★⏱ 8 min

Trigonometric identities

Learn and apply core trigonometric identities to simplify expressions and solve equations.

★★★★⏱ 9 min

Function properties: parity and periodicity

Analyze general function properties of even/odd parity and periodicity.

★★⏱ 5 min

Solving function equations and inequalities

Solve equations and inequalities involving all function types covered in this unit.

★★★⏱ 8 min

2. Common Pitfalls

Wrong move:

Forgetting to check the domain of composite or inverse functions

Why:

Most students only calculate the expression for the new function and ignore restricted domains, leading to incorrect full answers

Correct move:

Always state the full domain for any new function you create via composition or inversion

Wrong move:

Mixing up the order of horizontal transformations for $f (k x + c)$

Why:

Order of operations for input transformations is often reversed, leading to incorrect shifts or stretches

Correct move:

Factor out the coefficient of $x$ to get $f (k (x + c / k))$ , apply stretch/compression first, then shift

Wrong move:

Ignoring domain constraints when solving logarithmic or rational equations

Why:

This leads to extraneous solutions that do not satisfy the original equation

Correct move:

Always check all solutions against the domain of the original function before finalizing your answer

3. Quick Reference Cheatsheet

Concept / Property	Key Rule / Formula
Inverse function	$f (f^{- 1} (x)) = x$ , graph is reflection of $f (x)$ over $y = x$
General function transformation	$a f (b (x - h)) + k$ : $a$ = vertical stretch, $b$ = horizontal stretch, $h$ = horizontal shift, $k$ = vertical shift
Composite function	$(f \circ g) (x) = f (g (x))$ , domain is $x \in d o m (g)$ where $g (x) \in d o m (f)$
Logarithm core rules	$lo g (ab) = lo g a + lo g b$ , $lo g (a / b) = lo g a - lo g b$ , $lo g a^{b} = b lo g a$
Pythagorean identity	$sin^{2} θ + cos^{2} θ = 1$
Exponential-log conversion	$a^{x} = b ⟺ x = lo g_{a} b = \frac{l n b}{l n a}$
Even / odd functions	Even: $f (- x) = f (x)$ (symmetric over y-axis); Odd: $f (- x) = - f (x)$ (symmetric over origin)
Remainder Theorem	Remainder of $p (x)$ divided by $(x - a)$ equals $p (a)$

What's Next

Begin your learning of this unit with the first sub-topic on function definitions and notation, to build a strong foundational understanding. Once you complete all sub-topics in this Functions unit, you can progress to the first sub-topic of the next unit: Geometry and Trigonometry.