Composite and Inverse Functions Introductory
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Composite and Inverse Functions Introductory — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionEvaluating Composite FunctionsAssessment Practice
2 marks~3 minCriterion A
A surveyor uses a 3-4-5 right triangle to set a reference angle θ\theta, where θ\theta is the angle opposite the side of length 3.

The functions f(x)=sin⁡(x°)f(x) = \sin(x°) and g(x)=x+30g(x) = x + 30 model how the surveyor adjusts bearing angles.

Calculate the value of f(g(θ))f(g(\theta)), giving your answer to 3 significant figures. [2]
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Solutions

2QuestionChecking if Two Functions Are InversesAssessment Practice
3 marks~5 minCriterion C
The graph shows h(x)=12x+2h(x) = \frac{1}{2}x + 2 (solid line), the line y=xy = x (dashed), and three points claimed to belong to a function p(x)p(x): (0, −4)(0,\,-4), (2, 0)(2,\,0), and (4, 4)(4,\,4).
a
Calculate the coordinates of the points on h(x)h(x) at x=0x = 0, x=2x = 2, and x=4x = 4. [1]
b
Deduce the coordinates that the inverse of h(x)h(x) must pass through by reflecting your points from (a) across the line y=xy = x. [1]
c
Justify whether p(x)p(x) is the inverse of h(x)h(x), referring to your results from (b). [1]
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Solutions

3QuestionConcept of Inverse FunctionsAssessment Practice
6 marks~9 minCriterion D
A car's fuel efficiency is modelled by the function

f(x)=x12f(x) = \frac{x}{12}

where xx is the distance travelled in kilometres (km) and f(x)f(x) is the fuel used in litres (L). The model assumes constant driving conditions throughout any journey.
a
Determine the inverse function f−1(x)f^{-1}(x) and explain what it represents in this context. [2]
b
A driver has 30 L of fuel available. Using f−1(x)f^{-1}(x), calculate the maximum distance the driver can travel. [2]
c
The driver plans a 400 km motorway journey followed by a 20 km urban leg. Advise the driver whether 35 L of fuel is sufficient for the full journey, justifying your answer with reference to the model's assumption of constant driving conditions. [2]
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Solutions

4QuestionEvaluating Composite FunctionsAssessment Practice
6 marks~9 minCriterion B
A computer simulation models the spread of a digital signal through a network. At each step, the signal passes through two filters in sequence: filter ff doubles the strength, then filter gg adds 1 unit. Formally, f(x)=2xf(x) = 2x and g(x)=x+1g(x) = x + 1, giving composite function h(x)=g(f(x))h(x) = g(f(x)). The simulation begins at strength x0=1x_0 = 1 and updates by xn+1=h(xn)x_{n+1} = h(x_n).
a
Calculate x1x_1, x2x_2, x3x_3, and x4x_4. [2]
b
Deduce a general formula for xnx_n in terms of nn. [2]
c
The network has a maximum capacity of 1000 units. Justify whether this capacity is exceeded within the first 8 steps of the simulation. [2]

Solutions

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