Topic 2: Functions

Three 60-minute lesson plans on function notation, domain/range, and composite & inverse functions, pitched at DP AA (SL/HL), MYP Grade 10, and IGCSE Grade 10.

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A. DP Mathematics: Analysis & Approaches — SL/HL, Year 1

Caveat: content below is drawn from the pre-publication draft syllabus (first assessment 2029, Topic B: Functions). Verify against the final published guide once released.

Anchor concept: Function notation, domain and range, and composite and inverse functions (syllabus ref. B1). Taught in Year 1, immediately after Number & Algebra per SOW_DP1_AAHL.

Prior knowledge assumed: Sequences & series (Lesson 1 above, for notation fluency); solving equations; sketching simple graphs from MYP5/IGCSE.

Learning objectives

• SL+HL: Use function notation (f(x)), evaluate functions at a point, and determine domain and range from a graph or algebraic form.
• SL+HL: Form and evaluate composite functions (f\circ g(x)); find the inverse of a one-to-one function algebraically and graphically (reflection in (y=x)).
• HL only: State and justify the condition for (f\circ g) to exist (range of (g) must lie within domain of (f)); restrict the domain of a many-to-one function so that its inverse exists; identify odd/even functions algebraically and graphically.

Command terms used

Find (composite/inverse functions, domain/range); Determine (domain/range from a graph); State (a condition or restriction, HL); Justify (HL — why a restriction is needed for an inverse to exist).

Assessment focus (2029 draft AOs)

Abstraction — recognising that a real-world process (e.g. tax bands, unit conversion) can be modelled by composing two functions; Interpretation — explaining what (f^{-1}(a)) means in context.

ATL / skills focus

Thinking (moving between algebraic, graphical, and mapping-diagram representations of the same function); communication (correct use of function notation).

Starter (≈10 min)

Show a “function machine” diagram (input → operation → output) for two simple functions. Ask students to write each as (f(x)=), then predict what happens if the output of one feeds into the other.

Main teaching sequence (≈35 min)

1. Formalise function notation, domain, and range; work through 2–3 examples finding domain/range from both a graph and an algebraic rule (e.g. a rational function with an excluded value — previews Lesson on functions with asymptotes).
2. Composite functions: connect back to the starter’s chained function machine; formalise (f\circ g(x) = f(g(x))) and practise evaluating in both orders, showing order generally matters.
3. Inverse functions: derive algebraically (swap (x) and (y), solve for (y)) and confirm graphically via reflection in (y=x) using GDC.
4. HL-only extension (last ~10 min): existence condition for composites (range of inner ⊆ domain of outer), and domain-restriction for invertibility using (f(x)=x^2) restricted to (x\ge0) as the running example; brief intro to testing odd/even algebraically ((f(-x)=\pm f(x))). SL students instead complete extended composite/inverse practice, including a context-based problem (e.g. currency conversion then a flat fee, and its inverse).

Formative assessment / check for understanding

Cold-call pairs of students to evaluate (f(g(2))) vs (g(f(2))) on mini whiteboards to confirm the “order matters” point landed before moving to inverses.

Plenary / exit ticket (≈10 min)

“If (f(x)=2x+3), find (f^{-1}(x)) and verify that (f(f^{-1}(5))=5).” HL students additionally state the domain restriction (if any) required for (f) to have an inverse.

Resources

GDC graphing to confirm inverse-as-reflection visually (see Use of GDC in 2026.pdf); Haese AA SL textbook chapter on Functions under IB DP Mathematics/PPT of SL Mathematics/.

Vertical link note

Builds on MYP5’s function-notation and domain/range work (below), adding composite/inverse functions and, at HL, formal existence conditions absent from both MYP and IGCSE.

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B. MYP Mathematics — Grade 10 (MYP Year 5)

Anchor concept: Function notation, domain and range, and linear & quadratic function families (Thinking with Models branch).

Prior knowledge assumed: Plotting graphs of linear and quadratic relationships; solving linear and quadratic equations.

Learning objectives

• Use function notation (f(x)) to describe a relationship and evaluate it at given values.
• State the domain and range of a function from its graph, including for real-life constrained contexts (e.g. a function that only makes sense for positive integers).
• Identify the effect of the parameters (a, b, c) on the graph of (f(x)=ax^2+bx+c), and compare linear vs quadratic function behaviour.

Assessment criteria addressed

Criterion A (Knowing and understanding) — correct application of function notation and evaluating/solving; Criterion D (Applying mathematics in real-life contexts) — modelling a real situation with a function and justifying the appropriate domain restriction and accuracy of the answer.

Command terms used

Identify, Describe, Apply, Justify (D-criterion — why a particular domain restriction makes sense in context, e.g. “number of items sold” cannot be negative or non-integer).

ATL / skills focus

Thinking skills (linking parameter changes to graphical transformations); communication (organising input–output relationships using function notation and tables).

Starter (≈10 min)

Real context: “A parking garage charges a flat $5 entry fee plus $2 per hour.” Students write this as a function, evaluate it for a few values, and discuss whether every real number is a sensible input.

Main teaching sequence (≈35 min)

1. Formalise function notation from the starter; introduce domain/range explicitly, contrasting the “pure maths” domain (all reals) with the “real context” domain (non-negative, or integer hours) — this is the D-criterion hook.
2. Move to quadratics: given (f(x)=ax^2+bx+c), use GDC to vary (a), (b), (c) and observe effects on shape/position; students record observations in a table (Criterion A/C link).
3. Contrast linear vs quadratic growth using a paired real-life example (e.g. simple vs compound interest, or a projectile height model) — reading off domain/range appropriate to the context.
4. Guided practice: students are given 2–3 function/context pairs and must state an appropriate domain and range, justifying each restriction in a sentence.

Formative assessment / check for understanding

Quick verbal check during GDC exploration: ask students to predict what happens to the graph before changing a parameter, then confirm.

Plenary / exit ticket (≈10 min)

“A company’s profit is modelled by (P(n) = -2n^2 + 40n - 50), where (n) is the number of units sold. State a sensible domain for (n) and justify your answer.” (Direct Criterion D task.)

Resources

GDC or graphing software for parameter exploration; real-context worksheets (pricing, projectile motion) drawn from existing MYP5 resources.

Vertical link note

Feeds directly into the DP B1 work above (composite/inverse functions build on this notation and domain/range fluency); extends IGCSE’s function-notation work (below) with parameter exploration and explicit context-justification.

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C. IGCSE Mathematics — Grade 10 (Extended tier)

Caveat: IGCSE content below follows standard Cambridge (0580) / Edexcel International GCSE (4MA1) Extended-tier syllabus content. No official board specification file was found on this machine — cross-check against the current specification before formal use.

Anchor concept: Function notation (f(x)), and composite and inverse functions.

Prior knowledge assumed: Substitution into expressions; rearranging formulae; solving linear equations.

Learning objectives

• Evaluate a function (f(x)) for given values of (x), including for negative values and fractions.
• Find the composite function (fg(x)) (Cambridge notation) or (f(g(x))) and evaluate it at a point.
• Find the inverse function (f^{-1}(x)) algebraically.

Assessment objectives addressed

AO1 (recall and apply standard techniques — evaluating and combining functions); AO2 (reasoning — recognising why composite order matters).

Command terms used

Calculate, Find, Write down, Show that — standard Cambridge/Edexcel command-term conventions.

Starter (≈10 min)

Give (f(x)=3x-2) and ask students to evaluate (f(1)), (f(-2)), (f(0)) rapidly, then ask: “what value of (x) gives (f(x)=10)?” — quietly previews inverse thinking without naming it.

Main teaching sequence (≈35 min)

1. Formalise function notation and practise direct evaluation, including for a second function (g(x)), building fluency before combining them.
2. Composite functions: define (fg(x)=f(g(x))), work through evaluating (fg(2)) vs (gf(2)) for the same pair of functions to make “order matters” concrete, then move to finding a composite as a simplified algebraic expression.
3. Inverse functions: model the algebraic method (let (y=f(x)), swap (x) and (y), rearrange for (y)) using the starter’s reverse-evaluation question as the motivating example.
4. Practice set mixing direct evaluation, composite evaluation/simplification, and inverse-finding — including one “show that (f^{-1}(f(x))=x)” verification question.

Formative assessment / check for understanding

Board check: students hold up answers to a rapid sequence of “evaluate (fg(3))” style questions before moving to the harder simplification/inverse work.

Plenary / exit ticket (≈10 min)

”(f(x)=2x+1) and (g(x)=x^2). Find (fg(x)) and (f^{-1}(x)).” Students self-check using substitution of a specific value into both original and simplified composite forms.

Resources

Past-paper style function questions (Extended tier); GDC for numerical verification of composite and inverse results.

Vertical link note

Provides the notation and composite/inverse mechanics that MYP5 extends into modelling/parameter work, and that DP AA formalises further with existence conditions and domain restriction (HL).