Unit Overview

Functions

IB Mathematics AA SL· 5 min read 📊 20-23% of total exam score

1. Unit at a Glance

Functions are the backbone of quantitative modeling in mathematics, and this unit builds sequentially from core definitions to applied analysis of the most common function types you will use throughout the course. We start with foundational definitions and notation, then build to advanced algebraic operations and graphical transformations, before diving into the specific properties of each major function type that appear on IB exams.

Below are all sub-topics in this unit, ordered to build your knowledge progressively:

Function concepts, domain, range and notation

Learn the formal definition of a function and how to calculate domain and range for any relation.

★⏱ 10 min

Composite and inverse functions

Combine functions into composites and find inverses both algebraically and graphically.

★★⏱ 12 min

Transformations of graphs of functions

Master shifts, stretches, reflections and combinations of transformations for any function graph.

★★⏱ 15 min

Linear functions and their graphs

Review and extend properties of linear functions, gradients, and their graphical representations.

★⏱ 8 min

Quadratic functions, roots and discriminant

Analyze quadratic functions and use the discriminant to determine the number of real roots.

★★⏱ 14 min

Reciprocal and rational functions

Explore properties of reciprocal and simple rational functions including asymptotes.

★★★⏱ 12 min

Exponential and logarithmic functions and graphs

Learn key properties, graphs, and the inverse relationship between exponential and logarithmic functions.

★★★⏱ 16 min

2. Common Pitfalls

Wrong move:

Forgetting to restrict the 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 and restrict the domain before calculating an inverse

Wrong move:

Mixing up horizontal and vertical transformation rules

Why:

Horizontal transformations follow the opposite of intuitive direction, while vertical transformations follow intuition

Correct move:

Remember $f (x + a)$ shifts left by $a$ units, while $f (x) + a$ shifts up by $a$ units

Wrong move:

Swapping the order of composition in $f (g (x))$

Why:

Function composition is not commutative, so order changes the final result

Correct move:

Always substitute the inner function ( $g$ ) into the outer function ( $f$ ) in the correct order

3. Quick Reference Cheatsheet

Concept / Formula	Key Description
$y = f (x)$ function notation	Relates an input $x$ to exactly one output $y$
Composite function: $(f \circ g) (x) = f (g (x))$	The output of $g$ becomes the input of $f$
Inverse function property: $f^{- 1} (f (x)) = x$	Graph of $f^{- 1} (x)$ is reflection of $f (x)$ over $y = x$
Quadratic discriminant: $Δ = b^{2} - 4 a c$	Determines number of real roots: $Δ > 0$ (2), $Δ = 0$ (1), $Δ < 0$ (0)
Transformation: $a f (x + b) + c$	Horizontal shift $- b$ , vertical stretch $a$ , vertical shift $c$
Exponential-log inverse relationship: $lo g_{a} (a^{x}) = x$ , $a^{l o g_{a} x} = x$	Logarithms are the inverse of exponential functions
Rational function asymptotes	Vertical asymptotes where denominator = 0, horizontal asymptotes for end behavior

What's Next

Begin with the first sub-topic below to build your foundational understanding of functions, which all subsequent topics in this unit rely on. Once you complete all sub-topics in this unit on functions, you will move on to the first sub-topic of the next unit on trigonometry.