Composite Functions
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Composite Functions — Free MYP4 Mathematics (Extended) Practice Questions

1Questionf(g(x)) conceptConcept Practice
2 marks~3 minCriterion A
A construction team is installing a ramp for accessibility compliance. The ramp forms a right triangle: the horizontal base measures (x+1)(x + 1) m and the vertical rise measures 2x2x m, where x=2x = 2.

Building regulations require that the ramp angle θ\theta (opposite the vertical rise) must not exceed 55°55° to be approved.
a
Calculate the value of θ\theta. [1]
b
Advise the construction team whether the ramp should be approved, justifying your answer using your result from part (a). [1]
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2QuestionEvaluate compositeAssessment Practice
2 marks~3 minCriterion D
A mobile phone plan charges a fixed monthly fee of 20 dollars plus 0.10 dollars per minute of usage. A tax of 10% is applied to the total bill. The total cost in dollars is modelled by

C(t)=1.1(20+0.10t)C(t) = 1.1(20 + 0.10t)

where tt is the number of minutes used in a month.
a
Explain why multiplying only the fixed fee of 20 dollars by 1.1, rather than the entire expression (20+0.10t)(20 + 0.10t), produces an understated total cost. [1]
b
Discuss one limitation of using this model to predict a customer's monthly bill. [1]
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3QuestionEvaluate compositeConcept Practice
2 marks~3 minCriterion C
A delivery company charges a handling fee then applies a distance multiplier. The handling fee function is f(x)=x+2f(x) = x + 2 and the distance multiplier is g(x)=3xg(x) = 3x, where xx is the base cost in dollars.
a
Calculate f(g(x))f(g(x)) and g(f(x))g(f(x)) for x=1,2,3,4x = 1, 2, 3, 4. [1]
b
A customer claims the order of operations does not affect the final charge. Justify whether this claim is correct, using your results from part (a). [1]

Solutions

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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