Functions & Formulae – IB Math AI HL Complete Study Guide
Course: IB Mathematics: Applications & Interpretation (AI) Higher Level | Topic: 2 – Functions | Level: HL
This study guide covers every sub-topic in IB Math AI HL Topic 2: Functions, including all official formula booklet entries, worked examples, exam-style tips, and common mistakes students make in Paper 1, Paper 2, and Paper 3. Work through each section carefully — understanding functions deeply is critical because they appear throughout the entire AI HL course.
📚 Table of Contents
- Linear Functions — Equations, Gradients & Intercepts
- Domain and Range
- Types of Functions: One-to-One & Many-to-One
- Composite Functions
- Inverse Functions
- Quadratic Functions & Axis of Symmetry
- Exponential & Logarithmic Functions
- Logistic Function (HL Only)
- Transformations of Functions
- Official Formula Booklet Summary – Topic 2
- Exam Tips & Common Mistakes
- Frequently Asked Questions
1. Linear Functions — Equations, Gradients & Intercepts
A linear function produces a straight-line graph and has the general form f(x) = mx + c. Linear functions are the foundation of Topic 2 and appear regularly in modelling contexts across the AI HL exam.
Three Forms of a Linear Equation
Standard Form: ax + by + d = 0
Point-Slope Form: y − y₁ = m(x − x₁)
Where m is the gradient (slope) and c is the y-intercept.
Gradient Formula
Perpendicular Bisector
The perpendicular bisector of a line segment passes through its midpoint and is perpendicular to it. To find it: (1) find the midpoint using the midpoint formula; (2) find the negative reciprocal of the original gradient; (3) apply point-slope form.
📘 Worked Example
Find the equation of the line passing through A(2, 3) and B(6, 11).
Step 1 – Gradient: m = (11 − 3)/(6 − 2) = 8/4 = 2
Step 2 – Equation: y − 3 = 2(x − 2) → y = 2x − 1
2. Domain and Range
The domain of a function is the complete set of all possible input values (x-values) for which the function is defined. The range is the complete set of all possible output values (y-values) the function can produce.
| Term | Meaning | Notation |
|---|---|---|
| Domain | Set of all valid inputs (x-values) | x ∈ ℝ, x > 0, {x : x ≥ 2}, etc. |
| Range | Set of all possible outputs (y-values) | f(x) ∈ ℝ, f(x) > 0, {y : y ≤ 5}, etc. |
Rules for Identifying the Domain
- Square roots: The expression under a square root must be ≥ 0, so set the expression ≥ 0 and solve.
- Fractions (rational functions): The denominator cannot equal zero, so exclude values that make it zero.
- Logarithms: The argument of a logarithm must be strictly greater than zero (log(x) requires x > 0).
- No restriction stated: If a domain is not specified, assume x ∈ ℝ (all real numbers).
📘 Worked Example
Find the domain and range of f(x) = √(x − 4).
Domain: x − 4 ≥ 0 → x ≥ 4 → Domain: x ∈ [4, ∞)
Range: Since square root outputs are always ≥ 0 → Range: f(x) ∈ [0, ∞)
3. Types of Functions: One-to-One & Many-to-One
Understanding function types is critical for determining whether an inverse exists and how to approach composite function problems on the IB exam.
| Type | Definition | Inverse Exists? | Example |
|---|---|---|---|
| One-to-One (Injective) | Each x-value maps to a unique y-value | ✅ Yes | f(x) = 3x + 2 |
| Many-to-One | Two or more x-values map to the same y-value | ❌ No (unless domain restricted) | f(x) = x² |
Use the horizontal line test: if any horizontal line crosses the graph more than once, the function is many-to-one and does not have an inverse on its natural domain.
4. Composite Functions
A composite function applies one function to the result of another. The notation (f ∘ g)(x) means "apply g first, then apply f to the result." Order matters — f ∘ g is generally not equal to g ∘ f.
Read as: "f of g of x" — apply g first, then f
Domain of a Composite Function
Finding the domain of f ∘ g requires two steps:
- Find the range of g, then restrict it to values that lie within the domain of f.
- The domain of f ∘ g is the set of x-values that produce this restricted range through g.
📘 Worked Example
Given f(x) = x² and g(x) = x + 3, find (f ∘ g)(x) and (g ∘ f)(x).
(f ∘ g)(x): f(g(x)) = f(x + 3) = (x + 3)² = x² + 6x + 9
(g ∘ f)(x): g(f(x)) = g(x²) = x² + 3
These are different — composition is not commutative.
📘 Domain of Composite — Worked Example
f(x) = √x (domain: x ≥ 0), g(x) = x − 5. Find the domain of (f ∘ g)(x).
(f ∘ g)(x) = f(g(x)) = √(x − 5)
For the square root to be defined: x − 5 ≥ 0 → Domain: x ≥ 5
5. Inverse Functions
The inverse function f⁻¹(x) reverses the effect of f(x). If f maps x to y, then f⁻¹ maps y back to x. Geometrically, the graph of f⁻¹ is the reflection of f in the line y = x.
How to Find an Inverse Function (Algebraically)
- Replace f(x) with y: write y = f(x).
- Swap x and y: write x = f(y).
- Rearrange to make y the subject.
- Replace y with f⁻¹(x).
Domain & Range Swap: Domain of f⁻¹ = Range of f Range of f⁻¹ = Domain of f
📘 Worked Example
Find the inverse of f(x) = 3x − 7.
Step 1: y = 3x − 7
Step 2: Swap x and y: x = 3y − 7
Step 3: Rearrange: 3y = x + 7 → y = (x + 7)/3
Result: f⁻¹(x) = (x + 7)/3
Restricted Domain for Many-to-One Functions
For functions like f(x) = x² that are many-to-one, restrict the domain first (typically using the vertex as the boundary) so the function becomes one-to-one. Then find the inverse on this restricted domain. Remember: restricting the domain differently gives a different inverse.
6. Quadratic Functions & Axis of Symmetry
Quadratic functions have the standard form f(x) = ax² + bx + c (where a ≠ 0) and produce a parabola-shaped graph. The sign of a determines whether the parabola opens upward (a > 0) or downward (a < 0).
Three Forms of a Quadratic
| Form | Formula | Best Used For |
|---|---|---|
| Standard Form | f(x) = ax² + bx + c | Finding y-intercept (c), discriminant |
| Vertex Form | f(x) = a(x − h)² + k | Finding vertex (h, k) directly |
| Factored Form | f(x) = a(x − p)(x − q) | Finding x-intercepts (roots) p and q |
Key Formulae
Quadratic Formula — Solutions of ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / (2a), a ≠ 0
Discriminant: Δ = b² − 4ac
Discriminant Interpretation
- Δ > 0: Two distinct real roots (parabola crosses x-axis twice)
- Δ = 0: One repeated real root (parabola touches x-axis once — at the vertex)
- Δ < 0: No real roots (parabola does not cross x-axis); roots are complex
📘 Worked Example
f(x) = 2x² − 8x + 6. Find the axis of symmetry, vertex, and roots.
Axis of symmetry: x = −(−8)/(2 × 2) = 8/4 = 2
Vertex: f(2) = 2(4) − 8(2) + 6 = 8 − 16 + 6 = −2 → Vertex: (2, −2)
Discriminant: Δ = 64 − 4(2)(6) = 64 − 48 = 16 > 0 → two real roots
Roots: x = (8 ± √16)/4 = (8 ± 4)/4 → x = 3 or x = 1
7. Exponential & Logarithmic Functions
Exponential functions model rapid growth or decay and appear frequently in AI HL modelling problems. Logarithmic functions are their inverses.
Exponential Function
Natural Exponential: f(x) = eˣ
Exponential Growth Model: N(t) = N₀ · eᵏᵗ (k > 0 for growth, k < 0 for decay)
Logarithmic Function
Laws of Logarithms (AHL 1.9): logₐ(xy) = logₐx + logₐy logₐ(x/y) = logₐx − logₐy logₐ(xᵐ) = m · logₐx
Change of Base: logₐ(b) = ln(b) / ln(a)
The graph of y = logₐ(x) has a vertical asymptote at x = 0, passes through (1, 0), and is only defined for x > 0. It is the reflection of y = aˣ in the line y = x.
8. Logistic Function (HL Only)
The logistic function is an HL-only formula that models bounded growth — situations where a quantity grows rapidly at first but levels off at a maximum value known as the carrying capacity (L). It is one of the most important functions in the AI HL modelling context.
Parameters: L = carrying capacity (upper horizontal asymptote) k = growth rate constant C = constant related to initial conditions
Key Features of the Logistic Curve
| Feature | Value / Location |
|---|---|
| Upper horizontal asymptote | y = L (carrying capacity) |
| Lower horizontal asymptote | y = 0 |
| y-intercept | f(0) = L / (1 + C) |
| Point of inflection | Where f(x) = L/2 (halfway to carrying capacity) |
| Range | 0 < f(x) < L |
Real-World Applications
- Population growth limited by food supply or habitat
- Spread of disease (epidemiological models)
- Adoption of technology in a market
- Fish populations in a bounded ecosystem
📘 Worked Example
The population of a town is modelled by f(t) = 50000 / (1 + 24e⁻⁰·³ᵗ), where t is in years.
Carrying capacity: L = 50,000
Initial population (t = 0): f(0) = 50000 / (1 + 24) = 50000/25 = 2,000
Point of inflection: When population = 25,000 (i.e., L/2)
9. Transformations of Functions
Transformations alter the graph of a function systematically. Mastering transformations helps you quickly sketch graphs and interpret function behaviour — a frequent skill in IB AI HL exam questions.
| Transformation | Form | Effect on Graph |
|---|---|---|
| Vertical Translation | f(x) + a | Shifts graph up (a > 0) or down (a < 0) |
| Horizontal Translation | f(x − a) | Shifts graph right (a > 0) or left (a < 0) |
| Vertical Stretch/Compression | a · f(x) | Stretches vertically if |a| > 1; compresses if 0 < |a| < 1 |
| Horizontal Stretch/Compression | f(ax) | Compresses horizontally if |a| > 1; stretches if 0 < |a| < 1 |
| Reflection in x-axis | −f(x) | Flips graph upside down |
| Reflection in y-axis | f(−x) | Flips graph left–right |
| Reflection in y = x | f⁻¹(x) | Produces the inverse function graph |
10. Official Formula Booklet Summary – Topic 2
These are all the Topic 2 formulas listed in the IB Math AI HL official formula booklet. You must know what each formula means and when to apply it.
| Syllabus Ref | Description | Formula |
|---|---|---|
| SL 2.1 | Equation of a straight line | y = mx + c ; ax + by + d = 0 ; y − y₁ = m(x − x₁) |
| SL 2.1 | Gradient formula | m = (y₂ − y₁)/(x₂ − x₁) |
| SL 2.6 | Axis of symmetry of a quadratic | x = −b/(2a) |
| AHL 2.9 | Logistic function (HL only) | f(x) = L / (1 + Ce⁻ᵏˣ), L, k, C > 0 |
| Prior Learning (HL) | Quadratic formula | x = (−b ± √(b² − 4ac)) / (2a), a ≠ 0 |
11. Exam Tips & Common Mistakes
Students who score 6s and 7s in IB Math AI HL don't just know the formulas — they apply them accurately under time pressure and present working clearly. Here are the most impactful strategies for Topic 2.
Top 10 Tips for Functions in IB AI HL
- Always state the domain and range when sketching a function — marks are often awarded specifically for these.
- Show the reflection line y = x on any graph involving inverse functions for full marks.
- Label asymptotes explicitly on all exponential, logarithmic, and logistic graphs.
- Check function types before finding an inverse — if the function is many-to-one, state a restriction first.
- Verify composites by substituting a number. If (f ∘ g)(2) ≠ f(g(2)), you've made an error.
- State context when using logistic models. Don't just write L = 50000 — say "the carrying capacity is 50,000 people."
- For Paper 1 (no GDC), focus on algebraic manipulation and exact values. Know log laws by heart.
- For Paper 2 (GDC allowed), use regression tools to fit logistic and exponential models from data tables.
- Check discriminant before finding roots — it saves time to know whether to expect real solutions.
- Read transformation notation carefully — f(x − 3) + 2 involves both a horizontal shift AND a vertical shift. Apply each step separately.
Marking Scheme Awareness
IB marking schemes commonly award marks for: (M) Method, (A) Accuracy, (R) Reasoning, and (ft) Follow-through. Even if you make an arithmetic error early on, show all working to collect method marks for subsequent correct steps.
12. Frequently Asked Questions (FAQ)
What is the difference between IB Math AI HL and AA HL in Topic 2: Functions?
IB Math AI HL (Applications and Interpretation) focuses on applied, real-world modelling functions such as the logistic function and regression models. IB Math AA HL (Analysis and Approaches) goes deeper into analytical function theory, including rational functions, odd/even functions, and more complex graph analysis. AI HL emphasises using GDC technology to model real-world scenarios, while AA HL emphasises deeper algebraic and analytical treatment.
Is the logistic function given in the IB AI HL formula booklet?
Yes. The logistic function f(x) = L / (1 + Ce⁻ᵏˣ) is listed in the official IB Math AI HL formula booklet under AHL 2.9. You will have access to this formula in the exam, but you must understand what L, k, and C represent and how to use the GDC to fit logistic models to data.
Can every function have an inverse?
No. Only one-to-one (injective) functions have an inverse on their natural domain. Many-to-one functions like f(x) = x² or f(x) = sin(x) require a domain restriction to make them one-to-one before an inverse can be defined. The horizontal line test is a quick visual check: if any horizontal line crosses the graph more than once, the function is not one-to-one.
How do I find the domain of a composite function f ∘ g?
To find the domain of (f ∘ g)(x): first determine the range of g, then restrict these output values to those that fall within the domain of f. The domain of (f ∘ g) is the set of x-values from the domain of g that produce outputs lying within the domain of f.
What does the axis of symmetry formula give me for a quadratic?
The axis of symmetry formula x = −b/(2a) gives the x-coordinate of the vertex (turning point) of the parabola. Substitute this x-value back into f(x) to find the y-coordinate of the vertex. The vertex is the minimum point if a > 0 and the maximum point if a < 0.
