āš”

mathwithcye

KSSM Additional Mathematics Form 4 ā€¢ DLP

Chapter 1: Functions 1.2 Composite Functions
šŸŽ“
Cadet Mathematician Form 4 Alpha
ā­ 0 XP
Lesson Mastery Progress (6 Stages) 15% Complete
Stage A ā€¢ Mission Brief

Connecting Mathematical Machines in Series

Discover how chaining two mathematical operations models multi-stage real-world phenomena.

Real-World Case Study

The Expanding Oil Slick Pipeline

An offshore ship suffers an oil leak. The leaked oil spreads across the ocean surface forming a circle whose area is given by A(r) = πr2.

Environmental sensors detect that the radius r expands steadily with elapsed time t in hours: r(t) = 100t.

šŸ’” The Engineering Question: Rather than calculating the radius r every time, can we build a direct single-step formula that computes the contaminated surface area A directly from time t?

⇒ Composite Function: Ar(t) = π(100t)2 = 10 000πt2 m2!

šŸŽÆ Malaysian KSSM DSKP Learning Standards:

  • 1.2.1: Describe the outcome of composition of two functions f and g.
  • 1.2.2: Determine composite functions fg, gf, f2, g2.
  • 1.2.3: Determine the numerical image or object of a composite function.
  • 1.2.4: Understand the composite function existence condition (Range g ⊆ Domain f).
  • 1.2.5: Determine an unknown function when given the composite function and one component function.
  • 1.2.6: Solve authentic industrial and physical problems using composite functions.
Estimated time: 5ā€“8 mins
Stage B ā€¢ Discovery Lab

Interactive Digital Manipulatives

Experiment with sequential function pipelines, 3-set mapping diagrams, and mystery function solvers.

Select Pipeline Order:
Adjust Input Value (x): 3.0
Component Functions (TB p. 14 Ex 8):
  • f(x) = 2x (Doubler Machine)
  • g(x) = x2 − 5 (Square & Shift Machine)

āš” Crucial SPM Law: Notice how swapping the pipeline order produces entirely different numerical outputs! In general, fg(x) ≠ gf(x).

Select Domain Element in Set X:
Textbook Mapping Model (TB p. 13):

ā€¢ g : x → x2 (maps Set X to Set Y)
ā€¢ f : y → y + 2 (maps Set Y to Set Z)
ā€¢ fg : x → x2 + 2 (direct composite mapping)

šŸ’” Notice: The output of function g acts as the input object for function f.

Select Function Pair Scenario:
Condition for Existence of Composite Function:

The composite function fg exists if and only if every output of the inner function g is an acceptable input for the outer function f:

⇒ Range(g) ⊆ Domain(f)

Select Unknown Function Case:
Solving Strategy Guide:

ā€¢ Type 1: Form equation f[g(x)] = ... and isolate g(x) directly.
ā€¢ Type 2: Substitute u = g(x), express x in terms of u, then replace u back with x!

Stage C ā€¢ Interactive Worked Examples

Official Textbook Solutions Step-by-Step

Master official KSSM SPM methods through interactive intermediate checkpoints.

Stage D ā€¢ Guided Practice

Scaffolded Textbook Exercises

Solve authentic textbook exercises with instant multi-tier hints and feedback.

Stage E ā€¢ Independent Challenge

Tiered Challenge Bank & SPM Mastery

Test your composite function problem-solving skills across Foundation, Intermediate, and KBAT tiers.

Stage F ā€¢ Exit Ticket & Diagnostic Report

Concept Diagnostic Assessment

Answer 3 core diagnostic questions to generate your verified student completion certificate.