2.7 COMPOSITE & INVERSE FUNCTIONS

AHL 2.7: COMPOSITE & INVERSE FUNCTIONS

Concept Key Idea Core Skill
Composite Functions Apply one function inside another Correct order + domain checks
Inverse Functions Undo a function’s action Algebraic rearrangement + restriction

📌 Composite Functions in Context

  • Notation: (f ∘ g)(x) = f(g(x))
  • Meaning: Always apply g first, then apply f to the result.
  • Order matters: f ∘ g ≠ g ∘ f in general → composition is not commutative.
  • Domain rule: x must be valid for g, and g(x) must lie inside the domain of f.
  • Conceptually: Composite functions model multi-stage processes.

File:Composite Function Box.PNG - Wikimedia Commons

Composite_Function_Box.PNG

🔢 GDC & Technology Integration

  • Use your GDC to compute compositions directly by defining f(x) and g(x), then evaluating f(g(x)).
  • When finding inverses, use the GDC to verify correctness by checking that (f ∘ f⁻¹)(x) = x numerically.
  • For quadratic inverses with restrictions, graph both f and f⁻¹ to visually confirm mirror symmetry across y = x.
  • Always record the algebraic working — calculator output alone is not awarded full method marks.

Worked Example:
Let f(x) = 2x + 1 and g(x) = x² − 3.

  • (f ∘ g)(x) = f(g(x)) = f(x² − 3) = 2(x² − 3) + 1 = 2x² − 6 + 1 = 2x² − 5
  • (g ∘ f)(x) = g(2x + 1) = (2x + 1)² − 3 = 4x² + 4x − 2

This confirms that changing order changes the function entirely.

🌍 Real-World Connection

Composite functions appear in pricing chains (cost → tax → discount),
unit conversion pipelines, and physics (displacement → velocity → acceleration).
Each stage depends strictly on the previous output.

📌 Inverse Functions & Domain Restriction

  • Definition: f⁻¹ reverses the operation of f.
  • Identity Property: (f ∘ f⁻¹)(x) = x and (f⁻¹ ∘ f)(x) = x.
  • One-to-one requirement: Only functions that pass the horizontal line test can have an inverse.
  • If not one-to-one: We apply domain restriction.

📝 Paper Strategy

  • Always write compositions in the correct inside → outside order: f(g(x)), never f(x)g(x).
  • When finding inverses, you must explicitly show the swap of x and y before solving.
  • If the function is not one-to-one, you must state the domain restriction clearly or you lose accuracy marks.
  • Final answers must be labelled properly as f⁻¹(x), not just “y = …”.

IB Example:
f(x) = (x − 3)² − 2

  • This parabola fails the horizontal line test.
  • We restrict the domain to either x ≥ 3 or x ≤ 3.
  • Only after restriction does an inverse exist.

📌 Finding an Inverse Function (Algebraic Method)

  • 1. Replace f(x) with y
  • 2. Swap x and y
  • 3. Solve for y
  • 4. Replace y with f⁻¹(x)

Worked Example:
f(x) = 3x − 4

  1. y = 3x − 4
  2. x = 3y − 4
  3. x + 4 = 3y
  4. y = (x + 4)/3

∴ f⁻¹(x) = (x + 4)/3

📌 Composition with Inverses

If f and f⁻¹ are truly inverses:

  • (f ∘ f⁻¹)(x) = x → cancels completely
  • (f⁻¹ ∘ f)(x) = x → also cancels completely

If composition does not simplify to x, the inverse is incorrect.

📝 Paper Strategy

  • Always state the restricted domain with your inverse.
  • Verify inverses using (f ∘ f⁻¹)(x).
  • Never assume composition order can be swapped.
  • Method marks come from clean algebra — not calculator output.