Composite and Inverse Functions Introductory
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Composite and Inverse Functions Introductory — Free MYP4 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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2QuestionChecking if Two Functions Are InversesAssessment Practice
3 marks~5 minCriterion C
The graph shows f(x)=x+1f(x) = \sqrt{x+1} (solid curve), the line y=xy = x (dashed), and a dotted curve that is the reflection of ff across y=xy = x.
a
Explain the reflection property that connects a function to its inverse, as illustrated in this graph. [1]
b
Show that the inverse of f(x)=x+1f(x) = \sqrt{x+1} is g(x)=x2−1g(x) = x^2 - 1 with domain x≥0x \geq 0. [1]
c
The dotted curve is proposed as a model for a physical process that only produces non-negative outputs. Assess whether g(x)=x2−1g(x) = x^2 - 1 with domain x≥0x \geq 0 is an appropriate model for this constraint. [1]
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3QuestionChecking if Two Functions Are InversesAssessment Practice
8 marks~12 minCriterion D
A pharmaceutical company stores vaccines at temperatures between 2∘C2^\circ\text{C} and 8∘C8^\circ\text{C}. Their international storage log records only integer Fahrenheit temperatures. The conversion formulas are C=59(F−32)C = \dfrac{5}{9}(F - 32) and F=95C+32F = \dfrac{9}{5}C + 32.
a
Show that the two conversion formulas are inverse functions by computing both compositions C(F(C))C(F(C)) and F(C(F))F(C(F)). Clearly show each algebraic step. [3]
b
Explain how recording only integer Fahrenheit values introduces error in the corresponding Celsius temperatures. Support your explanation with a numerical example. [2]
c
The storage log shows Fahrenheit readings ranging from 35∘F35^\circ\text{F} to 46∘F46^\circ\text{F}. Convert these endpoints to Celsius, then advise the company whether this Fahrenheit range should be accepted as sufficient to guarantee that vaccines remain within the required 2∘C2^\circ\text{C} to 8∘C8^\circ\text{C} safe storage range. [3]
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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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