Composite Functions
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Composite Functions — Free MYP5 Mathematics (Standard) Practice Questions

1QuestionEvaluate compositeConcept Practice
2 marks~3 minCriterion A
The mapping diagram below shows how the functions f(x)f(x) and g(x)g(x) map values. Find the value of g(f(3))g(f(3)).
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2QuestionInterpret compositeAssessment Practice
2 marks~3 minCriterion D
A toy factory models the painting cost CC (in dollars) of a car as a function of its surface area AA (in cm²), and the surface area as a function of a single dimension dd (in cm), such that A(d)=6d2A(d) = 6d^2 and C(A)=0.05AC(A) = 0.05A.
a
Explain how the composite function C(A(d))C(A(d)) models the total painting cost in terms of dimension dd, showing the chain of dependence d→A→Cd \rightarrow A \rightarrow C. [1]
b
Interpret what a doubling of dimension dd means for the painting cost, and justify whether this composite model is sufficient for a factory manager to use when pricing larger car models. [1]
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3QuestionEvaluate compositeAssessment Practice
4 marks~6 minCriterion C
The graphs of two functions, f(x)f(x) and g(x)g(x), are shown below.

[The image should show two graphs on the same coordinate plane. f(x)f(x) is a curve passing through points (0, 2), (1, 3), (2, 4), (3, 3), and (4, 2). g(x)g(x) is a straight line passing through points (0, 1), (1, 2), (2, 3), and (3, 4).]

Explain how you would use the graphs to estimate the value of f(g(2))f(g(2)). Show your reasoning and state the approximate final value.
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4QuestionEvaluate compositeAssessment Practice
8 marks~12 minCriterion B
A computer simulation models the spread of a notification through a network. At each step, every active node activates exactly one new node and remains active. The number of active nodes at step nn is modelled by f(x)=2x+1f(x) = 2x + 1, applied repeatedly from x1=1x_1 = 1. The sequence is defined by xk+1=f(xk)x_{k+1} = f(x_k) for all positive integers kk.
a
Calculate the first five terms of the sequence. [2]
b
Deduce a formula for xnx_n in terms of nn, showing clearly how the pattern in part (a) leads to your general expression. [2]
c
Prove by mathematical induction that your formula holds for all positive integers nn. [2]
d
The network has a capacity of 1000 active nodes. Using your formula, advise the network administrator at which step they should intervene to prevent the capacity being exceeded, justifying your answer with reference to the simulation. [2]

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