Unit 2 · Topic 2: Functions
The Complete IB Mathematics: Analysis and Approaches Guide · IB Mathematics: Analysis and ApproachesUnit 2 · Topic 2: Functions
- The Complete IB Mathematics: Analysis and Approaches Guide
- IB Mathematics: Analysis and Approaches
- 5 sections
International Baccalaureate does not publish a per-unit weighting for this exam. Exam structure (official): SL: Paper 1 (90 min, 40%) · Paper 2 (90 min, 40%) · Internal Assessment (20%); HL: Paper 1 (120 min, 30%) · Paper 2 (120 min, 30%) · Paper 3 (60 min, 2 questions, 20%) · Internal Assessment (20%). Be ready to carry the same unit skills and representations into a complete answer on every paper. A function links inputs to single outputs; a composite function follows the inner rule first, while an inverse reverses the mapping only on a one-to-one domain. Rational function features come from numerator zeros, denominator zeros, and long-run behavior.
- How IB Mathematics: Analysis and Approaches assesses this International Baccalaureate publishes no per-unit percentage for IB Mathematics: Analysis and Approaches; Paper 1 and Paper 2 each account for 40% of the SL total and 30% of the HL total. · SL: Paper 1 (90 min, 40%) · Paper 2 (90 min, 40%) · Internal Assessment (20%) | HL: Paper 1 (120 min, 30%) · Paper 2 (120 min, 30%) · Paper 3 (60 min, 2 questions, 20%) · Internal Assessment (20%) · show the model with a function rule with domain and range, a transformed graph with intercepts and asymptotes, an exact factorization or bounded numerical root
- Key skills Straight Lines, Domain and Range, Graphs and Their Key Features, Composite, Identity, and Inverse Functions, Quadratic Functions, Equations, Inequalities, and the Discriminant, Rational, Exponential, and Logarithmic Functions and Solving Equations, Transformations of Graphs, Polynomial Functions, Factor and Remainder Theorems, Sum and Product of Roots (HL), Further Rational Functions, Odd and Even Functions, and Self-Inverse Functions (HL), Inequalities, Modulus Functions, and Further Graph Transformations (HL)
- How to study for Unit 2 This page turns a function rule with domain and range, a transformed graph with intercepts and asymptotes, an exact factorization or bounded numerical root into one route: State the domain and decide whether algebra, a graph, composition, inversion, or a numerical solver is the clearest route..
- The organizing decision Choose a representation that exposes the function's valid inputs, transformations, and key features.
What IB Mathematics: Analysis and Approaches Unit 2 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Straight Lines, Domain and Range, Graphs and Their Key Features
domain; range; intercept02Composite, Identity, and Inverse Functions
composite function; inverse function; restricted domain03Quadratic Functions, Equations, Inequalities, and the Discriminant
quadratic function; discriminant; factorization04Rational, Exponential, and Logarithmic Functions and Solving Equations
vertical asymptote; horizontal asymptote; numerical root05Transformations of Graphs
translation; vertical stretch; reflection06Polynomial Functions, Factor and Remainder Theorems, Sum and Product of Roots (HL)
polynomial; factor theorem; remainder theorem07Further Rational Functions, Odd and Even Functions, and Self-Inverse Functions (HL)
rational function; odd function; self-inverse function08Inequalities, Modulus Functions, and Further Graph Transformations (HL)
modulus inequality; closed interval; boundary pointInternational Baccalaureate does not publish a per-unit weighting for this exam.
Official unit name and weighting: IB subject guide / subject brief.
Representations, transformations, and valid inputs
Connect the published share to the unit model
A function links inputs to single outputs; a composite function follows the inner rule first, while an inverse reverses the mapping only on a one-to-one domain. Rational function features come from numerator zeros, denominator zeros, and long-run behavior.
A translation changes coordinates, a stretch changes distance from an axis, and an absolute value can reflect negative outputs. Modulus equation and modulus inequality work must preserve the distance interpretation and test interval boundaries.
Exact algebra is usually the clean route when factorization or inverse structure is visible. A graphic display calculator is useful for numerical intersections, but the setup, viewing region, and accepted root still belong in written work.
The decision that organizes this unit
Define the system and choose the route before calculating
Choose a representation that exposes the function's valid inputs, transformations, and key features.
State the domain and decide whether algebra, a graph, composition, inversion, or a numerical solver is the clearest route.
Mechanism route and repair branches
Relationships to preserve
- A composite function applies the inner function before the outer function.
- An inverse reverses a one-to-one mapping on a stated domain.
- Zeros, excluded inputs, and long-run behavior determine rational-function features.
Representations to read
- a function rule with domain and range
- a transformed graph with intercepts and asymptotes
- an exact factorization or bounded numerical root
Branches to reject
- treating inverse notation as a reciprocal
- giving a solver value without a search interval or substitution check
- forgetting that a domain restriction selects the inverse branch
| Key concept | Why it's hard | What scores |
|---|---|---|
| Function features | A graph can hide a hole or restricted input | States domain, intercepts, asymptotes, and transformations |
| Composition and inverse | Order matters and inverse notation is not a reciprocal | Shows the inner substitution and verifies a reversed pair |
| Modulus and numerical roots | Squaring can add invalid candidates | Splits cases or checks solver roots in the original equation |
How IB Mathematics: Analysis and Approaches assesses Topic 2: Functions
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| Straight Lines, Domain and Range, Graphs and Their Key Features | domain; range; intercept | treating inverse notation as a reciprocal |
| Composite, Identity, and Inverse Functions | composite function; inverse function; restricted domain | giving a solver value without a search interval or substitution check |
| Quadratic Functions, Equations, Inequalities, and the Discriminant | quadratic function; discriminant; factorization | forgetting that a domain restriction selects the inverse branch |
| Rational, Exponential, and Logarithmic Functions and Solving Equations | vertical asymptote; horizontal asymptote; numerical root | treating inverse notation as a reciprocal |
| Transformations of Graphs | translation; vertical stretch; reflection | giving a solver value without a search interval or substitution check |
| Polynomial Functions, Factor and Remainder Theorems, Sum and Product of Roots (HL) | polynomial; factor theorem; remainder theorem | forgetting that a domain restriction selects the inverse branch |
| Further Rational Functions, Odd and Even Functions, and Self-Inverse Functions (HL) | rational function; odd function; self-inverse function | treating inverse notation as a reciprocal |
| Inequalities, Modulus Functions, and Further Graph Transformations (HL) | modulus inequality; closed interval; boundary point | giving a solver value without a search interval or substitution check |
Transform and analyze a rational function
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Read rational-function features from algebra
Let f(x)=(x+1)/(x−2).
State (a) the vertical and horizontal asymptotes, (b) the intercepts, and (c) the transformation from y=1/x.
Show your working.
A. f(x)=1+3/(x−2), giving asymptotes x=2 and y=1, intercepts (−1,0) and (0,−1/2), and a right 2, vertical stretch 3, up 1 transformation.
✅ Correct! 😊 Correct. The transformed reciprocal form exposes both shifts and the stretch, while the numerator and f(0) give the intercepts.
B. f(x)=1−3/(x−2), giving asymptotes x=2 and y=1 and intercepts (1,0) and (0,1/2).
❌ Not quite 😭 The division sign is wrong: x+1=(x−2)+3. This error changes both intercepts and reflects the reciprocal term incorrectly.
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C. f(x)=1+3/(x+2), giving asymptotes x=−2 and y=1 and a shift 2 units left.
❌ Not quite 😭 The original denominator is zero at x=2, not x=−2. Changing x−2 to x+2 changes the function rather than rewriting it.
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D. The asymptotes are x=−1 and y=−1/2 because they are the x- and y-intercepts.
❌ Not quite 😭 Intercepts are points where the graph meets an axis; they are not asymptotes. The denominator and long-run value determine the two asymptotes.
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Full explanation
- Step 1Divide to rewrite (x+1)/(x−2)=1+3/(x−2).
- Step 2The denominator gives vertical asymptote x=2, and the translated reciprocal tends to horizontal asymptote y=1.
- Step 3The transformation is 2 units right, a vertical stretch by factor 3, then 1 unit up.
- Step 4Set x+1=0 for (−1,0), and evaluate f(0)=−1/2 for (0,−1/2).
Practice 2: Choose the correct inverse branch
Let h(x)=(x−2)² for x≥2, and let g(x)=x+1.
(a) Find (h∘g)(3). (b) Find h⁻¹, including its domain. (c) Verify the inverse using the value from part (a).
Show your working.
A. (h∘g)(3)=4; h⁻¹(y)=2+√y for y≥0; h⁻¹(4)=4 and h(4)=4.
✅ Correct! 😊 Correct. Applying g first gives 4, the restriction selects the nonnegative square-root branch, and the two values verify the reversed mapping.
B. (h∘g)(3)=2; h⁻¹(y)=2−√y for y≥0.
❌ Not quite 😭 This does not apply g before h and chooses outputs below 2. That branch contradicts the stated domain x≥2.
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C. (h∘g)(3)=4; h⁻¹(y)=1/(y−2)² for y≠2.
❌ Not quite 😭 The composition value is right, but inverse notation does not mean reciprocal. The inverse must undo squaring on the restricted branch.
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D. (h∘g)(3)=16; h⁻¹(y)=2±√y for y≥0.
❌ Not quite 😭 Here h is applied to 4 as (4−2)²=4, not 16. Keeping both square-root branches also prevents the inverse from being a function.
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Full explanation
- Step 1Apply the inner function first: g(3)=4, then h(4)=(4−2)²=4.
- Step 2From y=(x−2)² and x≥2, take the nonnegative branch: x=2+√y.
- Step 3The range of h is y≥0, so this is the domain of h⁻¹.
- Step 4Check h⁻¹(4)=4 and h(4)=4, which reverses the mapped pair.
Key terms for Unit 2: Topic 2: Functions
Models, uses, and boundaries
- Rational features
- The rational function can be written as 1 plus 3 divided by x minus 2. Use it to read the transformations and both asymptotes.
- Restricted inverse
- On the restricted domain, the inverse output is 1 plus the nonnegative square root of the input. Use it when the original quadratic is restricted to inputs at least 1.
- Modulus distance
- The distance of 2x minus 3 from zero is at most 5. Use it to convert the modulus inequality into one double inequality.
- Numerical root
- The numerical target is the real zero of x cubed minus x minus 1. Use it with a bounded interval and a substitution check.
- Exact roots
- The quadratic factors into x minus 2 times x minus 3. Use it when exact factorization is visible and technology is unnecessary.
IB Mathematics: Analysis and Approaches Unit 2 FAQ
How much of IB Mathematics: Analysis and Approaches does Unit 2 carry?
International Baccalaureate does not publish a per-unit weighting for this exam.
What is the first move on a Topic 2: Functions problem?
State the domain and decide whether algebra, a graph, composition, inversion, or a numerical solver is the clearest route.
Which relationships should I preserve?
A composite function applies the inner function before the outer function. An inverse reverses a one-to-one mapping on a stated domain. Zeros, excluded inputs, and long-run behavior determine rational-function features.
Which representations should I practice?
Practice moving among a function rule with domain and range, a transformed graph with intercepts and asymptotes, an exact factorization or bounded numerical root.
What error should I check before submitting an answer?
Check for treating inverse notation as a reciprocal; giving a solver value without a search interval or substitution check; forgetting that a domain restriction selects the inverse branch.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Transform and analyze a rational function
- Compose a function and verify its inverse
- Solve a modulus inequality as a distance
- Choose an analytic or numerical root route
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related IB Mathematics: Analysis and Approaches unit guides
IB Mathematics: Analysis and Approaches Exam Guide & Review
The whole exam and its official unit sequence.01Topic 1: Number and Algebra
International Baccalaureate does not publish a per-unit weighting for this exam.03Topic 3: Geometry and Trigonometry
International Baccalaureate does not publish a per-unit weighting for this exam.04Topic 4: Statistics and Probability
International Baccalaureate does not publish a per-unit weighting for this exam.05Topic 5: Calculus
International Baccalaureate does not publish a per-unit weighting for this exam.How to study IB Mathematics: Analysis and Approaches Unit 2
Start with the organizing decision
Before solving, restate the decision in operational terms: Choose a representation that exposes the function's valid inputs, transformations, and key features.. Your first written move should be to State the domain and decide whether algebra, a graph, composition, inversion, or a numerical solver is the clearest route..
Practice the same idea in several representations
Rotate through a function rule with domain and range, a transformed graph with intercepts and asymptotes, an exact factorization or bounded numerical root. Use each representation to practice Straight Lines, Domain and Range, Graphs and Their Key Features, Composite, Identity, and Inverse Functions, Quadratic Functions, Equations, Inequalities, and the Discriminant, Rational, Exponential, and Logarithmic Functions and Solving Equations, Transformations of Graphs, Polynomial Functions, Factor and Remainder Theorems, Sum and Product of Roots (HL), Further Rational Functions, Odd and Even Functions, and Self-Inverse Functions (HL), Inequalities, Modulus Functions, and Further Graph Transformations (HL), and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for treating inverse notation as a reciprocal; giving a solver value without a search interval or substitution check; forgetting that a domain restriction selects the inverse branch. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official IB subject guide / subject brief for the 2027 administration.
Where to go next
Return to the IB Mathematics: Analysis and Approaches to see every unit and plan your revision.