Functions — Free IB Maths AA HL Practice Questions

1FoundationSAQ-SReal-world applications of functions (e.g. growth models)s8 marksPaper 1~12 min

Criterion: Find Marks: 8 A small town has a population of

12\,500

people. A demographer models the population

t

years from now using the linear function

P(t) = 12500 + 340t

(a)

Write down the values of the initial population and the annual rate of change, and state what each represents in context. [2 marks]

(b)

Determine the number of complete years it will take for the population to first exceed

15\,000

people. [2 marks]

(c)

The demographer notes that a neighbouring town has population

Q(t) = 18000 - 120t

. Determine the year in which the two towns have equal populations, and evaluate whether a linear model is likely to remain valid for both towns beyond this point. [4 marks]

Solutions

2MasterySAQ-SReal-world applications of functions (e.g. growth models)s5 marksPaper 1~8 min

A biologist is studying the population of a species of bird on a remote island. The population size

P

(in hundreds) is modelled by

P(t) = 12 + 3\ln(t+1)

, where

t

is the time in years since the study began.

(a)

Show that the population first reaches 2700 birds after

T = e^5 - 1

years. [2 marks]

(b)

Find the average rate of change of

P

, in hundreds of birds per year, over the interval

0 \leq t \leq T

, giving your answer in exact form. [2 marks]

(c)

The biologist claims that the rate of growth of the population is always decreasing. Determine whether this claim is correct, justifying your answer with a mathematical argument. [1 mark]

Solutions

3ChallengeSAQ-LReal-world applications of functions (e.g. growth models)s8 marksPaper 1~12 min

A pharmaceutical company is testing a new drug. The concentration

C(t)

(in mg/L) of the drug in a patient's bloodstream at time

t

(in hours) after administration is modelled by

C(t) = \frac{at}{t^2 + b}, \quad t \geq 0,

where

a

and

b

are positive constants.

(a)

Calculate the value of

t

at which the maximum concentration occurs, expressing your answer in terms of

b

. [3 marks]

(b)

The maximum concentration is

C_{\max} = 5\,\text{mg/L}

, occurring at

t = 2

hours. Determine the values of

a

and

b

. [3 marks]

(c)

A drug is considered therapeutically effective while

C(t) \geq 1\,\text{mg/L}

. Using your values from (b), determine the total duration for which the drug remains therapeutically effective. [2 marks]

Solutions

4FoundationSAQ-SReal-world applications of functions (e.g. growth models)s10 marksPaper 1~15 min

A museum models its daily visitor numbers by

V(t) = 800 + 200\sin\!\left(\frac{\pi t}{12}\right)

where

t

is the number of hours after the museum opens at 9:00 am and

0 \leq t \leq 12

.

(a)

State the number of visitors when the museum opens. [1 mark]

(b)

Find the number of visitors at 12:00 pm (

t = 3

). [2 marks]

(c)

Find the maximum number of visitors predicted by this model and the time at which this maximum occurs. [2 marks]

(d)

Find the values of

t

for which the number of visitors is less than

900

, giving your answer as an interval. Total: 10 marks [5 marks]

Solutions

5FoundationSAQ-SDefinition and types of functions (one-to-one, onto etc.)s5 marksPaper 1~8 min

The function

f: \mathbb{R} \to \mathbb{R}

is defined by

f(x) = \begin{cases} 2x + 1 & x \geq 0 \\ x^2 + 1 & x < 0 \end{cases}

-

(a)

Show that

f

is injective (one-to-one). - [3 marks]

(b)

Determine whether

f

is surjective (onto). Justify your answer. [2 marks]

Solutions

6MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)s5 marksPaper 1~8 min

A function

h: \mathbb{R} \to \mathbb{R}

is defined by

h(x) = \begin{cases} x+1, & x \leq 2 \\ 5-x, & x > 2 \end{cases}

(a)

Show that

h

is not a one-to-one function. [2 marks]

(b)

Show that

h(x) \leq 3

for all

x \in \mathbb{R}

, and hence show that

h

is not an onto function. [2 marks]

(c)

Deduce the range of

h

, justifying that every value in your stated range is achieved. [1 mark]

Solutions

7ChallengeSAQ-LDefinition and types of functions (one-to-one, onto etc.)s8 marksPaper 1~12 min

A function

f: \mathbb{R} \to \mathbb{R}

is defined by

f(x) = \ln(x^2 + 1) + 2

.

(a)

Determine whether

f

is injective (one-to-one). Justify your answer. [3 marks]

(b)

Determine whether

f

is surjective (onto). Justify your answer. [3 marks]

(c)

State a restricted domain

D \subseteq \mathbb{R}

and a restricted codomain

C \subseteq \mathbb{R}

such that

f: D \to C

is a bijection. Hence write down an expression for

f^{-1}(x)

, stating its domain. [2 marks]

Solutions

8FoundationSAQ-SDefinition and types of functions (one-to-one, onto etc.)s8 marksPaper 1~12 min

The function

g: \mathbb{R} \to \mathbb{R}

is defined by

g(x) = x^2 - 4x + 5

.

(a)

State whether

g

is a one-to-one function. Give a reason for your answer. [2 marks]

(b)

Find the range of

g

. [2 marks]

(c)

The domain of

g

is restricted to

x \geq k

, where

k \in \mathbb{R}

, so that the resulting function

h

has an inverse

h^{-1}

. State the smallest value of

k

and find

h^{-1}(x)

, stating its domain. [4 marks]

Solutions

9FoundationSAQ-STranslation, reflection, stretching, and compressions5 marksPaper 1~8 min

The function

f

is defined by

f(x) = \sqrt{x}

for

x \geq 0

. The graph of

f

is reflected in the

y

-axis and then translated

2

units to the right and

1

unit up to obtain the graph of

g

.

(a)

Find an expression for

g(x)

, stating its domain. [3 marks]

(b)

The function

h

is defined by

h(x) = g(x)

for

x \leq k

. Determine the value of

k

such that

h^{-1}

exists, and find an expression for

h^{-1}(x)

. [2 marks]

Solutions

10MasterySAQ-STranslation, reflection, stretching, and compressions7 marksPaper 1~11 min

The function

f

is defined by

f(x) = x^2 - 4x + 3

, for

x \in \mathbb{R}

.

(a)

The graph of

f

is translated by the vector

\begin{pmatrix} 2 \\ -1 \end{pmatrix}

to give the graph of a function

g

. Show that

g(x) = x^2 - 8x + 14

. [2 marks]

(b)

The graph of

g

is reflected in the

y

-axis to give the graph of a function

h

. Find an expression for

h(x)

in the form

h(x) = ax^2 + bx + c

. [2 marks]

(c)

Find the minimum value of

h

and hence determine whether the graphs of

f

and

h

intersect. [3 marks]

Solutions

11ChallengeSAQ-LTranslation, reflection, stretching, and compressions8 marksPaper 1~12 min

The below shows the graph of

y = f(x)

, where

f

is a quadratic function. The graph has a vertex at

(2, -1)

and passes through the point

(0, 3)

.

(a)

Determine the equation of

f

in the form

f(x) = a(x - h)^2 + k

. [2 marks]

(b)

The graph of

f

is transformed by: - a reflection in the

x

-axis, - followed by a horizontal stretch with scale factor

2

, - followed by a translation by the vector

\begin{pmatrix} 1 \\ 4 \end{pmatrix}

. The resulting function is

g

. Determine the coordinates of the vertex of the graph of

g

. [3 marks]

(c)

Write

g(x)

in the form

g(x) = px^2 + qx + r

, where

p, q, r \in \mathbb{R}

, and hence find the exact

x

-coordinates of the points where the graph of

g

intersects the

x

-axis. [3 marks]

Solutions

12FoundationSAQ-STranslation, reflection, stretching, and compressions5 marksPaper 1~8 min

The graph of

y = f(x)

undergoes the following transformations in order: 1. A vertical compression by a factor of

\dfrac{1}{3}

2. A reflection in the

x

-axis 3. A horizontal translation of

4

units to the left 4. A vertical translation of

2

units upward The resulting graph is

y = g(x)

.

(a)

Given

f(x) = x^2

, find

g(x)

in the form

g(x) = a(x+h)^2 + k

. [3 marks]

(b)

The graph of

g

intersects the

x

-axis at two points. Find the exact

x

-coordinates of these points. [2 marks]

Solutions

13FoundationSAQ-SSine, cosine, and tangent functionss8 marksPaper 1~12 min

The height,

h

metres, of the water above a fixed point a harbour entrance is modelled by

h(t) = 3 + 4\sin\!\left(\frac{\pi t}{6}\right)

where

t

is the time in hours after midnight,

0 \leq t \leq 24

.

(a)

State the amplitude of

h(t)

. [1 mark]

(b)

Calculate the height of the water at 2:00 am. [2 marks]

(c)

A boat requires a minimum water height of

5\,\text{m}

to enter the harbour safely. Determine the values of

t

, in the interval

0 \leq t \leq 12

, for which the boat may enter the harbour. (Give your answers as exact values.) [5 marks]

Solutions

14MasterySAQ-STrigonometric identities and equations5 marksPaper 1~8 min

Consider the equation

2\cos^2 x + 3\sin x = 3

, for

0 \leq x \leq 2\pi

.

(a)

Show that this equation can be written as

2\sin^2 x - 3\sin x + 1 = 0

. [2 marks]

(b)

Hence solve

2\cos^2 x + 3\sin x = 3

for

0 \leq x \leq 2\pi

, giving your answers in exact form. [3 marks]

Solutions

15ChallengeSAQ-LSine, cosine, and tangent functionss8 marksPaper 1~12 min

Consider the function

f(x) = \sin x + \cos x

defined for

x \in \mathbb{R}

.

(a)

Show that

f(x)

can be written in the form

R\sin(x + \alpha)

, where

R > 0

and

0 \leq \alpha < 2\pi

, and determine the exact values of

R

and

\alpha

. [3 marks]

(b)

Hence, determine the exact solutions of the equation

f(x) = 1

for

0 \leq x \leq 2\pi

. [2 marks]

(c)

The function

g

is defined by

g(x) = \dfrac{1}{f(x)}

for

x \in [0, 2\pi]

, excluding values where

f(x) = 0

. State the

x

-values in

[0, 2\pi]

where

g(x)

is undefined, and hence determine the range of

g

on the largest open interval within

[0, 2\pi]

on which

g

is continuous. [3 marks]

Solutions

16FoundationSAQ-SSine, cosine, and tangent functionss6 marksPaper 1~9 min

Consider the function

f(x) = \cos x

, for

0 \leq x \leq \pi

.

(a)

State the range of

f

. [1 mark]

(b)

The function

g

is defined by

g(x) = 2f(x)^2 - 1

. (i) Show that

g(x) = \cos(2x)

. [2]
(ii) Hence determine the value of

x \in \left[0, \pi\right]

for which

g(x) = f(x)

, giving your answer in exact form. [3 marks]

Solutions

17MasterySAQ-SReal-world applications of functions (e.g. growth models)s6 marksPaper 2~9 min

A biologist studies two bacterial colonies growing in a laboratory. Both colonies start with a mass of

2.4

grams at

t = 0

, where

t

is the time in hours. Colony A is modelled by

M(t) = 2.4 + 0.18t

. Colony B is modelled by

N(t) = 2.4 \times 1.12^{t}

.

(a)

Write down the rate of growth, in grams per hour, of Colony A. [1 mark]

(b)

Determine the value of

t

at which Colony A reaches a mass of

6

grams. [2 marks]

(c)

Calculate the value of

t

at which Colony B reaches a mass of

6

grams. Give your answer correct to 3 significant figures. [2 marks]

(d)

Using your answers to parts (b) and (c), explain which model predicts a greater mass at

t = 15

hours, and justify your answer. [1 mark]

Solutions

18MasterySAQ-SReal-world applications of functions (e.g. growth models)s9 marksPaper 2~14 min

A small business sells handmade crafts. The number of crafts sold per week,

C

, is modelled by a linear function of the number of weeks since the business opened,

w

. After 2 weeks, 15 crafts were sold. After 6 weeks, 35 crafts were sold.

(a)

Calculate the linear function

C(w)

in the form

C(w) = a + bw

. [3 marks]

(b)

The owner also considers an exponential growth model

E(w) = 10 \times 1.25^w

. Determine the value of

w

at which the two models predict the same weekly sales. Give your answer correct to 3 significant figures. [3 marks]

(c)

For

w > 8

, determine which model predicts higher weekly sales, and calculate the difference in predicted sales at

w = 12

. Give your answer correct to 3 significant figures. [3 marks]

Solutions

19MasterySAQ-SReal-world applications of functions (e.g. growth models)s6 marksPaper 2~9 min

A biologist is studying the growth of two populations of bacteria in separate petri dishes. Dish A initially contains

500

bacteria and grows linearly at a constant rate of

40

bacteria per hour. Dish B initially contains

200

bacteria and grows exponentially according to the model

P_B(t) = 200 \times (1.05)^t

, where

t

is the time in hours.

(a)

State an expression for

P_A(t)

, the number of bacteria in Dish A after

t

hours. [1 mark]

(b)

Determine the value of

t

at which the populations in the two dishes are equal, giving your answer correct to three significant figures. [3 marks]

(c)

At the first integer hour after the time found in part (b), calculate the difference in the number of bacteria between the two dishes, giving your answer to the nearest whole number. [2 marks]

Solutions

20MasterySAQ-SSolving problems using functionss9 marksPaper 2~14 min

A company manufactures cylindrical containers with a fixed volume of

500\,\text{cm}^3

. The cost of the materials for the curved surface is

\text{USD}\,0.02

per

\text{cm}^2

, and the cost for the two circular ends is

\text{USD}\,0.03

per

\text{cm}^2

.

(a)

Show that the total cost,

C

dollars, of manufacturing one container can be expressed as

C = \frac{20}{r} + 0.06\pi r^2

where

r\,\text{cm}

is the radius of the container. [3 marks]

(b)

Calculate the value of

r

that minimises the total cost. Give your answer correct to 3 significant figures. [3 marks]

(c)

A rival manufacturer claims that using a radius of

r = 4.00\,\text{cm}

produces a cost within 5 percent of the minimum. Determine whether this claim is correct, justifying your answer with calculations. [3 marks]

Solutions

21MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)s6 marksPaper 2~9 min

The function

g

is defined by

g(x) = \sqrt{9 - x^2}

for

x \in \mathbb{R}

, with its natural domain.

(a)

State the domain and range of

g

. [2 marks]

(b)

Determine whether

g

is one-to-one. Justify your answer. [2 marks]

(c)

The function

h

is defined by

h(x) = \sqrt{9 - x^2}

for

x \in [0, 3]

, with codomain

[0, 3]

. Show that

h

is a bijection (both one-to-one and onto). [2 marks]

Solutions

22MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)s6 marksPaper 2~9 min

The function

f

is defined by

f(x) = \dfrac{1}{x-1} + 2

for

x \in \mathbb{R},\ x \neq 1

.

(a)

Determine the range of

f

. [2 marks]

(b)

Find an expression for

f^{-1}(x)

, stating its domain. [2 marks]

(c)

The function

g

is defined by

g(x) = \dfrac{x+3}{x-1}

for

x \in \mathbb{R},\ x \neq 1

. Show that

f^{-1} \circ f^{-1}(x) = g(x)

. [2 marks]

Solutions

23MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)s6 marksPaper 2~9 min

Consider the function

f: \mathbb{R} \to \mathbb{R}

defined by

f(x) = x^3 - 3x

.

(a)

Determine whether

f

is a one-to-one function. Justify your answer. [2 marks]

(b)

The function

g

is defined by restricting

f

to the domain

[1, \infty)

with codomain

[-2, \infty)

. Show that

g

is a bijection. [2 marks]

(c)

Hence find an expression for

g^{-1}(x)

, stating its domain. [2 marks]

Solutions

24MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)s6 marksPaper 2~9 min

A function

f

is defined by

f(x) = \dfrac{3x - 5}{x - 2}

,

x \in \mathbb{R}

,

x \neq 2

.

(a)

Show that

f

is a one-to-one function its natural domain. [2 marks]

(b)

The function

f

is now restricted to the domain

D = (2, +\infty)

. (i) Write down the range of

f

on the domain

D

. [1]
(ii) Find an expression for

f^{-1}(x)

, stating its domain. [3 marks]

Solutions

25MasterySAQ-STranslation, reflection, stretching, and compressions6 marksPaper 2~9 min

The function

f

is defined by

f(x) = x^2 + 4x + 3

, for

x \in \mathbb{R}

.

(a)

Write

f(x)

in the form

(x+h)^2 + k

, where

h, k \in \mathbb{Z}

. [2 marks]

(b)

The graph of

f

is transformed to the graph of

g

by a horizontal translation of

3

units to the left, followed by a vertical stretch with scale factor

2

about the

x

-axis. Find an expression for

g(x)

in the form

ax^2 + bx + c

, where

a, b, c \in \mathbb{Z}

. [3 marks]

(c)

Hence, write down the range of

g

. [1 mark]

Solutions

26MasterySAQ-STranslation, reflection, stretching, and compressions6 marksPaper 2~9 min

The graph of

y = f(x)

passes through the points

A(-1,\,0)

,

B(0,\,2)

, and

C(2,\,0)

. The function

g

is obtained by applying the following transformations to

f

in order: - a horizontal stretch with scale factor

2

, - a reflection in the

x

-axis, - a vertical translation of

3

units upwards.

(a)

Determine the coordinates of the images of

A

,

B

, and

C

under the combined transformation. [3 marks]

(b)

Given that

f(x) = -\dfrac{1}{2}x^2 + \dfrac{1}{2}x + 2

, find an expression for

g(x)

and hence find the

x

-coordinates of the points where the graph of

g

crosses the

x

-axis. [3 marks]

Solutions

27MasterySAQ-STranslation, reflection, stretching, and compressions6 marksPaper 2~9 min

The graph of

y = \cos x

, for

0 \leq x \leq 2\pi

, is transformed to the graph of

y = h(x)

by the following sequence of transformations: 1. A horizontal stretch with scale factor

\dfrac{1}{2}

2. A translation by

\begin{pmatrix} \dfrac{\pi}{3} \\[4pt] 0 \end{pmatrix}

3. A vertical stretch with scale factor

\dfrac{2}{3}

4. A reflection in the

y

-axis

(a)

Determine an expression for

h(x)

in the form

h(x) = a\cos(bx + c) + d

. [3 marks]

(b)

Find the exact values of

x

in the interval

0 \leq x \leq 2\pi

for which

h(x) = \dfrac{1}{3}

. [3 marks]

Solutions

28MasterySAQ-SThe effect of transformations on the graph of a function s6 marksPaper 2~9 min

The function

f

is defined by

f(x) = x^2 - 4x + 7

, for

x \in \mathbb{R}

. The graph of

f

is transformed by: - a horizontal translation of

3

units to the right, - followed by a vertical stretch with scale factor

2

about the

x

-axis, - followed by a vertical translation of

5

units downwards. Let

g

be the resulting function.

(a)

Determine an expression for

g(x)

in the form

a(x - h)^2 + k

. [4 marks]

(b)

Hence, determine the range of

g

and explain why

g(x) = 0

has no real solutions. [2 marks]

Solutions

29MasterySAQ-SSine, cosine, and tangent functionss6 marksPaper 2~9 min

A Ferris wheel has a diameter of

50\,\text{m}

and completes one full revolution every

40

seconds. The lowest point of the wheel is

2\,\text{m}

above the ground. A passenger boards at the lowest point at time

t = 0

seconds. Let

h(t)

metres denote the height of the passenger above the ground at time

t

seconds.

(a)

Show that

h(t) = 27 - 25\cos\!\left(\dfrac{\pi}{20}t\right)

. [3 marks]

(b)

Calculate the first time after

t = 0

at which the passenger is at a height of

40\,\text{m}

above the ground. Give your answer correct to 3 significant figures. [3 marks]

Solutions

30MasterySAQ-SSine, cosine, and tangent functionss6 marksPaper 2~9 min

The depth of water in a harbour,

d

metres, can be modelled by the function

d(t) = 5 + 3\sin\!\left(\frac{\pi}{6}t\right) + 4\cos\!\left(\frac{\pi}{6}t\right),

where

t

is the time in hours after midnight.

(a)

Express

d(t)

in the form

d(t) = 5 + R\sin\!\left(\dfrac{\pi}{6}t + \alpha\right)

, where

R > 0

and

0 \leq \alpha < 2\pi

. State the value of

R

and give

\alpha

in radians correct to 3 significant figures. [3 marks]

(b)

Hence find the first time strictly after midnight at which the depth of water is exactly

9

metres. Give your answer in hours and minutes, correct to the nearest minute. [3 marks]

Solutions

31MasterySAQ-SSine, cosine, and tangent functionss6 marksPaper 2~9 min

A particle moves along a straight line. Its velocity

v\,\text{m s}^{-1}

at time

t

seconds is given by

v(t) = 6\cos\!\left(2t - \frac{\pi}{3}\right), \quad t \geq 0.

(a)

Write down the maximum speed of the particle. [1 mark]

(b)

Determine the first positive time at which the particle is at rest. Give your answer in exact form. [2 marks]

(c)

The displacement of the particle from its initial position is

s(t)

, where

s(0) = 0

. Find

s(t)

in the form

s(t) = a\sin\!\left(2t - \frac{\pi}{3}\right) + b,

where

a

and

b

are exact constants. [3 marks]

Solutions

32MasterySAQ-SSine, cosine, and tangent functionss8 marksPaper 2~12 min

A surveyor measures the angle of elevation to the top of a vertical tower from two points on level ground. From point A,

50\,\text{m}

from the base of the tower, the angle of elevation is

\theta

. From point B,

30\,\text{m}

from the base of the tower, the angle of elevation is

(\theta + 20)^\circ

. Let

h

metres be the height of the tower.

(a)

Show that

h = 50\tan\theta

and

h = 30\tan(\theta + 20)^\circ

. [2 marks]

(b)

Hence, calculate the value of

\theta

, giving your answer in degrees correct to

1

decimal place. [3 marks]

(c)

A third point C is located on the same level ground such that C is collinear with A and B, and the angle of elevation from C to the top of the tower is

\frac{\theta + (\theta + 20)^\circ}{2}

. Determine whether C is closer to A or to B, justifying your answer. [3 marks]

Solutions

33ChallengeLAQReal-world applications of functions (e.g. growth models)s12 marksPaper 3~18 min

A biologist studies the population growth of a rare bird species on an isolated island. The population

P

(in hundreds) at time

t

(in years) satisfies the logistic differential equation

\frac{dP}{dt} = \frac{1}{5}P\!\left(1 - \frac{P}{K}\right)

where

K

is the carrying capacity (in hundreds) and

P(0) = 1

.

(a)

Show that the general solution satisfying

P(0) = 1

is

P(t) = \frac{K}{1 + (K-1)e^{-t/5}}

[5 marks]

(b)

Given

K = 10

, calculate the time at which the population reaches

90\%

of the carrying capacity. [3 marks]

(c)

For the model with

K = 10

, the growth rate

\frac{dP}{dt}

is a function of

P

. (i) Determine the value of

P

at which

\frac{dP}{dt}

is maximised. [1 mark]
(ii) Hence analyse how the rate of population growth changes as

P

increases from

P = 5

to

P = 9

, and explain what this implies about the time taken for the population to grow from

50\%

to

90\%

of

K

compared to the time taken to grow from

10\%

to

50\%

of

K

. Total: 12 marks [3 marks]

Solutions

34ChallengeLAQReal-world applications of functions (e.g. growth models)s10 marksPaper 3~15 min

A pharmaceutical company is developing a new drug. Initially,

5000

bacteria are placed in a culture medium. The number of bacteria,

N(t)

, after

t

hours is recorded below.

t

(hours) —

0

—

3

—

10

N(t)

—

5000

—

7700

—

14000

(a)

Show that the three data points are consistent with the linear model

N(t) = 900t + 5000, \quad 0 \leq t \leq 10.

[3 marks]

(b)

Using the model from part (a), find the time

t

at which the bacterial population first doubles its initial size. [2 marks]

(c)

A researcher proposes that for

t > 10

, the population grows exponentially. Using the value

N(10) = 14000

as the initial condition and assuming the population reaches

N = 50000

at

t = 20

hours, determine the exponential model

N(t) = 14000\,e^{k(t-10)}

for

t > 10

, giving

k

correct to three significant figures. [2 marks]

(d)

Compare the predictions of the linear model and the exponential model at

t = 15

. Hence evaluate which model is more appropriate for

t > 10

, justifying your answer with reference to both the mathematical behaviour of each model and the biological context of bacterial growth. [3 marks]

Solutions

35ChallengeLAQDomain and range of functionss10 marksPaper 3~15 min

Consider the function

f(x) = \sqrt{\ln\!\left(\dfrac{x+1}{x-1}\right)}

, where

x \in \mathbb{R}

.

(a)

Determine the maximal domain of

f

, expressing your answer in exact form. [3 marks]

(b)

For the function

g(x) = f(x) + \sqrt{4 - x^2}

, find the domain of

g

. [2 marks]

(c)

Prove that the range of

f

is

[0, \infty)

. You must justify continuity and use a formal argument (such as the Intermediate Value Theorem or strict monotonicity) to establish that all values in

[0, \infty)

are attained. [5 marks]

Solutions

36ChallengeLAQDomain and range of functionss10 marksPaper 3~15 min

A function

h

is defined by

h(x) = \dfrac{2x^2 + 3x - 5}{x^2 - 4}

, where

x \in \mathbb{R}

.

(a)

Find the domain of

h

and state the equations of all vertical asymptotes, justifying why no removable discontinuities exist. [3 marks]

(b)

Determine the equation of the horizontal asymptote of

h

and state the behaviour of

h(x)

as

x \to 2^+

. [2 marks]

(c)

Prove that the range of

h

is

\mathbb{R}

. [5 marks]

Solutions

37ChallengeLAQTranslation, reflection, stretching, and compressions10 marksPaper 3~15 min

Data

1. A horizontal stretch by scale factor

\dfrac{1}{2}

(compression towards the

y

-axis). 2. A reflection in the

y

-axis. 3. A translation by

\begin{pmatrix} 3 \\ 0 \end{pmatrix}

.

Consider

f(x) = \ln(x)

for

x > 0

. Let

g(x)

be obtained by applying the following transformations to

f(x)

in the order

(a)

Show that

g(x) = \ln(-2x + 6)

, and state the domain of

g

. [4 marks]

(b)

Using the log identity

\ln(ab) = \ln a + \ln b

, express

g(x)

in the form

A + \ln(B - x)

, where

A

and

B

are exact constants to be found. [2 marks]

(c)

A student claims that applying the three transformations in the order: translate → horizontal stretch → reflect produces the same function

g(x)

. Determine the function

g_{\text{new}}(x)

produced by this reordering, state its domain, and hence evaluate whether the student's claim is correct, justifying your answer by comparing

g_{\text{new}}(x)

with

g(x)

from part (b). [4 marks]

Solutions

38ChallengeLAQTranslation, reflection, stretching, and compressions10 marksPaper 3~15 min

Starting from

f(x) = \cos x

, the following four transformations are applied in the order listed: 1. Vertical stretch by scale factor

a

, where

a > 0

2. Horizontal translation of

\dfrac{\pi}{3}

units to the left 3. Reflection in the

x

-axis 4. Vertical translation of

b

units downward

(a)

Show that the resulting function is

g(x) = -a\cos\!\left(x + \dfrac{\pi}{3}\right) - b

, and hence determine the values of

a

and

b

given that

g(x) = -2\cos\!\left(x + \dfrac{\pi}{3}\right) - 1

. [4 marks]

(b)

A new function

h(x)

is obtained by applying the same four transformations to

f(x) = \cos x

, but in reverse order: 1. Vertical translation of

b

units downward 2. Reflection in the

x

-axis 3. Horizontal translation of

\dfrac{\pi}{3}

units to the left 4. Vertical stretch by scale factor

a

Using

a = 2

and

b = 1

, derive

h(x)

in simplified form. [3 marks]

(c)

Using the identity

\cos(A + B) = \cos A \cos B - \sin A \sin B

, expand both

g(x)

and

h(x)

fully. Hence determine all values of

x

in

[0, 2\pi]

for which

g(x) = h(x)

. [3 marks]

Solutions

39ChallengeLAQSine, cosine, and tangent functionss10 marksPaper 3~15 min

The function

f

is defined by

f(x) = \sin x + \cos x

, for

x \in \mathbb{R}

.

(a)

Prove that

f(x)

can be written in the form

R\sin(x+\alpha)

, where

R > 0

and

\alpha \in \left[0,\dfrac{\pi}{2}\right]

, stating the exact values of

R

and

\alpha

. [4 marks]

(b)

Hence prove that the equation

f(x) = \dfrac{3}{2}

has no solutions. [3 marks]

(c)

Using your result from (a), determine the number of solutions of the equation

f(x) = k

in the interval

0 \leq x \leq 2\pi

for each possible value of the constant

k \in \mathbb{R}

. Justify your answer. [3 marks]

Solutions

40ChallengeLAQSine, cosine, and tangent functionss12 marksPaper 3~18 min

Consider the function

g(x) = \tan x - \sec x

, defined for

x \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)

.

(a)

Prove that

g(x) = \dfrac{\sin x - 1}{\cos x}

. [2 marks]

(b)

Using the result from part (a), prove that

g(x) = -\tan\!\left(\dfrac{\pi}{4} - \dfrac{x}{2}\right)

. [5 marks]

(c)

Using the result from part (b): (i) Find

\displaystyle\lim_{x \to \frac{\pi}{2}^{-}} g(x)

. [1]
(ii) Show that

g'(x) = \sec x(\sec x - \tan x)

, and hence determine

\displaystyle\lim_{x \to \frac{\pi}{2}^{-}} g'(x)

. [4 marks]

Solutions

Functions — Free Maths AA HL Practice Questions