Functions — Free IB Maths AI SL Practice Questions

1FoundationSAQ-SDefinition and types of functions (one-to-one, onto etc.)6 marksPaper 1~9 min

A function

f

is defined by

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

, for

x \neq 2

.

(a)

Write down the range of

f

. [1 mark]

(b)

Find

f^{-1}(x)

, stating its domain. [3 marks]

(c)

Hence determine the value of

x

for which

f(x) = f^{-1}(x)

, and explain why there is only one such value. [2 marks]

Solutions

2MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)5 marksPaper 1~8 min

Consider the function

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

, defined for

x \in \mathbb{R}

.

(a)

Show that

g

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

(b)

The domain of

g

is restricted to

x \geq k

so that

g

becomes one-to-one. Write down the smallest possible value of

k

. [1 mark]

(c)

Hence find the range of

g

for the restricted domain

x \geq k

. [2 marks]

Solutions

3ChallengeSAQ-LDefinition and types of functions (one-to-one, onto etc.)8 marksPaper 1~12 min

A function

f

is defined by

f(x) = \dfrac{2x+3}{x-1}

for

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

.

(a)

Determine whether

f

is a one-to-one function. [3 marks]

(b)

Write down the range of

f

. [1 mark]

(c)

A second function

g

is defined by

g(x) = \dfrac{ax+b}{cx+d}

, where

a,b,c,d \in \mathbb{R}

and

c \neq 0

. The graph of

g

has a vertical asymptote at

x = 2

and a horizontal asymptote at

y = 3

. The point

(0,\ 1)

lies on the graph of

g

. (i) Find the values of

a

,

b

,

c

, and

d

. [2]
(ii) The composite function

h = g \circ f

is defined on the domain of

f

. Find

h(x)

, simplifying your answer fully, and state its range. [2 marks]

Solutions

4FoundationSAQ-SDefinition and types of functions (one-to-one, onto etc.)5 marksPaper 1~8 min

The graph of

f

is shown below. The domain of

f

is

-4 \leq x \leq 4

. -

(a)

(i) State whether

f

is a one-to-one function. [1] -
(ii) Justify your answer to (a)(i). - [1 mark]

(b)

Find the range of

f

. [3 marks]

Solutions

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

A function

f

is defined by

f(x) = \sin x

, for

0 \leq x \leq \pi

.

(a)

Show that

f

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

(b)

The domain is restricted to

0 \leq x \leq a

so that

f

becomes one-to-one. Write down the maximum possible value of

a

. [1 mark]

(c)

Hence state the range of

f

for the restricted domain

0 \leq x \leq a

. [2 marks]

Solutions

6FoundationSAQ-STranslation, reflection, stretching and compression5 marksPaper 1~8 min

A function

f

is defined by

f(x) = x^2

for

x \in \mathbb{R}

.

(a)

The graph of

f

is translated by

\begin{pmatrix} 3 \\ -2 \end{pmatrix}

. Find the equation of the resulting function

g

, giving your answer in the form

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

. [2 marks]

(b)

The graph of

g

is then reflected in the

x

-axis to give the function

h

. Find the equation of

h

, giving your answer in the form

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

. [2 marks]

(c)

Write down the range of

h

. [1 mark]

Solutions

7MasterySAQ-STranslation, reflection, stretching and compression6 marksPaper 1~9 min

The function

f

is defined by

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

, for

x \in \mathbb{R}

. The graph of

f

is translated by the vector

\begin{pmatrix} 3 \\ -5 \end{pmatrix}

to obtain the graph of a function

g

.

(a)

Show that

g(x) = x^2 - 10x + 23

. [3 marks]

(b)

Write down the coordinates of the vertex of the graph of

g

. [1 mark]

(c)

Hence, state the range of

g

and explain why

g

does not have an inverse function for

x \in \mathbb{R}

. [2 marks]

Solutions

8ChallengeSAQ-LTranslation, reflection, stretching and compression8 marksPaper 1~12 min

The function

f

is defined by

f(x) = x^3 - 4x

for

x \in \mathbb{R}

. The function

g

is defined by

g(x) = -2f(x - 3) + 1

.

(a)

State the sequence of three transformations, in the correct order, that maps the graph of

f

onto the graph of

g

. [3 marks]

(b)

The point

P(-1,\, 3)

lies on the graph of

f

. Calculate the coordinates of the image of

P

on the graph of

g

. [2 marks]

(c)

(i) The

x

-intercepts of

f

occur where

f(x) = 0

. Explain why the images of these points under the transformation are not

x

-intercepts of

g

. [1]
(ii) Determine the exact coordinates of the three points on the graph of

g

that correspond to the

x

-intercepts of

f

. [2 marks]

Solutions

9FoundationSAQ-STranslation, reflection, stretching and compression5 marksPaper 1~8 min

The graph of

y = \sqrt{x}

for

x \geq 0

is transformed by a horizontal stretch with scale factor

\dfrac{1}{2}

, followed by a translation by

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

.

(a)

Find the equation of the resulting function, giving your answer in the form

y = a\sqrt{bx} + c

, where

a

,

b

, and

c

are integers. [3 marks]

(b)

State the range of the resulting function. [1 mark]

(c)

Find the coordinates of the point on the resulting graph where

x = 4

. [1 mark]

Solutions

10MasterySAQ-STranslation, reflection, stretching and compression5 marksPaper 1~8 min

The graph of

y = f(x)

passes through

(-2,\,0)

,

(0,\,2)

,

(1,\,0)

, and

(3,\,-2)

. The function

g

is obtained by reflecting the graph of

f

in the

y

-axis, followed by a vertical stretch with scale factor

2

.

(a)

Show that

g(1) = -4

. [3 marks]

(b)

The equation

g(x) = 2f(x)

is satisfied for a particular value of

x

visible on the graph. Determine this value of

x

and justify your answer. [2 marks]

Solutions

11FoundationSAQ-SSine, cosine and tangent functions6 marksPaper 1~9 min

The depth,

d

metres, of water in a harbour is modelled by

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

where

t

is the time in hours after midnight,

0 \leq t \leq 24

.

(a)

Write down the minimum depth of water in the harbour. [1 mark]

(b)

Find the depth of water at 2:00 AM. Give your answer correct to 3 significant figures. [2 marks]

(c)

Find the first time after midnight at which the depth of water is

3.5\,\text{m}

. Give your answer correct to the nearest minute. (Total: 6 marks) [3 marks]

Solutions

12MasterySAQ-SGraphing trigonometric functions5 marksPaper 1~8 min

The temperature

T

in degrees Celsius inside a greenhouse over a 24-hour period is modelled by

T(t) = 18 - 5\sin\!\left(\frac{\pi}{12}(t - 6)\right)

where

t

is the time in hours after midnight,

0 \leq t < 24

.

(a)

Show that the temperature at 14:00 is

13.7^\circ\text{C}

, correct to one decimal place. [2 marks]

(b)

Find the two times during the day when the temperature is exactly

15.5^\circ\text{C}

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

Solutions

13ChallengeSAQ-LTrigonometric identities and equations8 marksPaper 1~12 min

The depth of water,

d

metres, at a harbour entrance is modelled by

d(t) = 4.5 + 2.5\cos\!\left(\frac{\pi}{6}t\right)

where

t

is the time in hours after midnight.

(a)

A boat requires a minimum depth of

5\,\text{m}

to enter the harbour safely. Determine the two smallest positive values of

t

for which

d(t) = 5

. Give your answers correct to three significant figures. [4 marks]

(b)

A second boat requires a minimum depth of

6\,\text{m}

. Using your answers from part (a), determine the total length of time during

0 \leq t \leq 12

when both boats can safely enter the harbour simultaneously. Give your answer in hours correct to three significant figures. [4 marks]

Solutions

14FoundationSAQ-SGraphing trigonometric functions10 marksPaper 1~15 min

The depth of water,

d

metres, in a harbour is modelled by the function

d(t) = 3.2\sin\!\left(\frac{\pi}{6}t\right) + 4.8

where

t

is the time in hours after midnight.

(a)

State the amplitude and the period of

d(t)

. [2 marks]

(b)

Calculate the minimum depth of water in the harbour and state the first time after midnight at which this minimum occurs. *A vessel requires a minimum depth of

3.0\,\text{m}

to enter the harbour safely.* [3 marks]

(c)

Determine the values of

t

, in the interval

0 \leq t \leq 12

, at which the depth equals

3.0\,\text{m}

, and hence state the length of time during this interval for which the harbour is unsafe for the vessel. [5 marks]

Solutions

15MasterySAQ-SGraphing trigonometric functions5 marksPaper 1~8 min

The depth of water,

D

metres, in a harbour is modelled by the function

D(t) = 3.5\sin\!\left(\frac{\pi}{6}t\right) + 5

where

t

is the time in hours after midnight,

0 \leq t < 24

.

(a)

Write down the amplitude and period of

D(t)

. [2 marks]

(b)

Hence state the maximum and minimum depths of water in the harbour. [1 mark]

(c)

Find the total number of hours in the 24-hour period during which the depth of water is greater than

7\,\text{m}

. [2 marks]

Solutions

16MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)6 marksPaper 2~9 min

A small independent bookstore tracks the number of customers,

C

, each day. The owner models the number of customers by

C(t) = at^2 + bt + c

, where

t

is the number of hours after opening,

0 \leq t \leq 10

. The following data are recorded:

t

(hours) —

C

(customers)

0

—

5

2

—

21

5

—

75

(a)

Show that

C(t) = 2t^2 + 4t + 5

. [3 marks]

(b)

Determine whether

C(t)

is one-to-one on the domain

0 \leq t \leq 10

. Justify your answer using the vertex of the parabola and the behaviour of the function the domain. [3 marks]

Solutions

17MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)6 marksPaper 2~9 min

A function

h

is defined by

h(x) = \begin{cases} x^2 + 1, & x \leq 0 \\ 2x + 1, & x > 0 \end{cases}

(a)

Calculate

h(-2)

and

h(3)

. [2 marks]

(b)

Determine whether

h

is one-to-one. Justify your answer with a counterexample or algebraic reasoning. [2 marks]

(c)

State the range of

h

and hence determine whether

h

is onto when the codomain is

\mathbb{R}

. [2 marks]

Solutions

18MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)7 marksPaper 2~11 min

A function

f

is defined by

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

, for

x \geq 2

.

(a)

Calculate

f(6)

and find the value of

x

for which

f(x) = 5

. [2 marks]

(b)

Determine the range of

f

. [2 marks]

(c)

The function

g

is defined by

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

for

x > 2

. Show that

g(x) = \dfrac{1}{\sqrt{x-2}}

, and hence find

g^{-1}(x)

, stating its domain. [3 marks]

Solutions

19MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)10 marksPaper 2~15 min

A company manufactures cylindrical water tanks. The volume

V

(in

\text{m}^3

) of a tank is given by

V = \pi r^2 h

, where

r

is the radius (in m) and

h

is the height (in m). For a particular model, the height is fixed at

h = 3\,\text{m}

and the radius satisfies

0.5 \leq r \leq 2.5

.

(a)

Show that

V(r) = 3\pi r^2

and state the range of

V

on the given domain. [3 marks]

(b)

Determine whether

V(r)

is a one-to-one function

0.5 \leq r \leq 2.5

. Justify your answer. [2 marks]

(c)

Calculate the radius required for a tank with volume

20\,\text{m}^3

, correct to 3 significant figures. [2 marks]

(d)

A second model has the same fixed height of

3\,\text{m}

but the domain is extended to

-2.5 \leq r \leq 2.5

. Explain whether

V(r)

remains one-to-one on this new domain, and state the physical reason why this extended domain is not appropriate for modelling tank radius. [3 marks]

Solutions

20MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)6 marksPaper 2~9 min

A biologist is studying the population

P

of a species of bird on an island. The population is modelled by

P(t) = 200 + 80\sin\left(\frac{\pi t}{6}\right)

where

t

is the time in months after 1 January 2020, for

0 \leq t \leq 12

.

(a)

State the range of

P(t)

for the given domain. [2 marks]

(b)

Determine the total length of time, in months, during which the population exceeds 240 birds. Give your answer correct to 3 significant figures. [2 marks]

(c)

Calculate the first value of

t

, in months correct to 3 significant figures, at which the population reaches 260 birds. [2 marks]

Solutions

21MasterySAQ-SThe effect of transformations on the graph of a function6 marksPaper 2~9 min

The graph of

f(x) = x^2

is transformed to obtain the graph of

g(x) = (x-3)^2 + 2

.

(a)

State the two transformations that map the graph of

f

to the graph of

g

. [2 marks]

(b)

The point

P(2,\,4)

lies on the graph of

f

. Calculate the coordinates of the image of

P

on the graph of

g

. [2 marks]

(c)

A function

h

is obtained by applying the following two transformations to

f

in order: - translate by vector

\begin{pmatrix}-4\\0\end{pmatrix}

- then translate by vector

\begin{pmatrix}0\\5\end{pmatrix}

The graph of

h

is then reflected in the

y

-axis to give the graph of

k

. Determine the equation of

k

in the form

k(x) = (x+a)^2 + b

, where

a,b \in \mathbb{Z}

. [2 marks]

Solutions

22MasterySAQ-SThe effect of transformations on the graph of a function6 marksPaper 2~9 min

The graph of

f(x) = \sqrt{x}

for

x \geq 0

is transformed to the graph of

g(x) = \sqrt{x-2} + 1

.

(a)

State the domain of

g

. [1 mark]

(b)

The point

Q(9,\,3)

lies on the graph of

f

. Calculate the coordinates of the image of

Q

on the graph of

g

. [2 marks]

(c)

A function

k

is obtained by applying a single translation to

f

, such that

k(x) = \sqrt{x+c} + d

, where

c, d \in \mathbb{Z}

. The graph of

k

passes through the point

P(1,\,5)

and has a minimum value of

3

. Determine the values of

c

and

d

. [3 marks]

Solutions

23MasterySAQ-SComposition and inverse of functions6 marksPaper 2~9 min

A company models the daily profit,

P

dollars, from selling

x

units of a product using the function

P(x) = -0.1x^2 + 50x - 2000

, for

0 \leq x \leq 400

. The company also models the daily tax payable,

T

dollars, as a function of profit:

T(P) = 0.25P + 500

.

(a)

Find the composition

(T \circ P)(x)

, simplifying your answer. [2 marks]

(b)

Calculate the daily tax payable when

x = 150

units are sold. [2 marks]

(c)

Find the values of

x

for which the daily tax payable equals USD 1500, giving your answers correct to the nearest unit. [2 marks]

Solutions

24MasterySAQ-SThe effect of transformations on the graph of a function6 marksPaper 2~9 min

The graph of

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

for

x \neq 0

is transformed to the graph of

g(x) = \dfrac{1}{x+2} - 3

.

(a)

State the two transformations that map the graph of

f

to the graph of

g

. [2 marks]

(b)

The point

R(-1,\,-1)

lies on the graph of

f

. Calculate the coordinates of the image of

R

under the transformation from

f

to

g

. [2 marks]

(c)

The graph of

g

is translated by

p

units in the positive

x

-direction and

q

units in the positive

y

-direction to obtain the graph of

h

. The asymptotes of

h

are

x = 3

and

y = 1

. Determine the values of

p

and

q

. [2 marks]

Solutions

25MasterySAQ-SThe effect of transformations on the graph of a function6 marksPaper 2~9 min

The graph of

f(x) = — x —

is transformed to the graph of

g(x) = — x + 1 — - 4

.

(a)

State the two transformations that map the graph of

f

to the graph of

g

. [2 marks]

(b)

The point

S(-3,\, 3)

lies on the graph of

f

. Calculate the coordinates of the image of

S

on the graph of

g

. [2 marks]

(c)

The point

T'(5,\, -2)

lies on the graph of

g

and is the image of point

T

on the graph of

f

. Determine the coordinates of

T

. [2 marks]

Solutions

26MasterySAQ-STrigonometric identities and equations6 marksPaper 2~9 min

A particle moves along a straight line. Its displacement

s

(in metres) from a fixed point at time

t

(in seconds) is given by

s(t) = 3\sin(2t) - 4\cos(2t)

(a)

Show that

s(t)

can be written in the form

R\sin(2t - \alpha)

, where

R > 0

and

0 < \alpha < \dfrac{\pi}{2}

, and determine the values of

R

and

\alpha

. Give

\alpha

correct to three significant figures. [3 marks]

(b)

State the maximum displacement of the particle from the fixed point. [1 mark]

(c)

Calculate the smallest positive value of

t

for which

s(t) = 2

. Give your answer correct to three decimal places. [2 marks]

Solutions

27MasterySAQ-SSine, cosine and tangent functions7 marksPaper 2~11 min

A water wheel at an amusement park has a diameter of

12

metres. The centre of the wheel is

8

metres above the ground. The wheel rotates clockwise at a constant speed, completing one full revolution every

40

seconds. A passenger starts their ride at the lowest point of the wheel at time

t = 0

seconds.

(a)

Show that the height

h

, in metres, of the passenger above the ground can be modelled by

h(t) = 8 - 6\cos\!\left(\frac{\pi}{20}t\right)

where

t

is the time in seconds. [2 marks]

(b)

Calculate the height of the passenger above the ground after

15

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

(c)

Determine the first time after

t = 0

when the passenger is exactly

11

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

Solutions

28MasterySAQ-SSine, cosine and tangent functions6 marksPaper 2~9 min

A Ferris wheel has a radius of

15\,\text{m}

and its centre is

18\,\text{m}

above the ground. The wheel completes one full revolution every

90

seconds. A rider starts at the lowest point at

t = 0

.

(a)

Write down an expression for the height

h(t)

, in metres, of the rider above the ground in the form

h(t) = d - a\cos(bt)

, stating the values of

a

,

b

, and

d

. [1 mark]

(b)

Calculate the height of the rider after

1

minute and

15

seconds. Give your answer correct to

3

significant figures. [2 marks]

(c)

Determine the total time, in seconds, during the first

90

seconds that the rider is above

25\,\text{m}

. Give your answer correct to the nearest second. [3 marks]

Solutions

29MasterySAQ-SSine, cosine and tangent functions6 marksPaper 2~9 min

The temperature inside a greenhouse varies during the day and can be modelled by the function

T(t) = 22 + 6\sin\!\left(\frac{\pi}{12}(t-8)\right)

where

T

is the temperature in degrees Celsius and

t

is the time in hours after midnight.

(a)

Determine the maximum and minimum temperatures in the greenhouse. [2 marks]

(b)

Calculate the temperature at 14:30. Give your answer correct to 3 significant figures. [2 marks]

(c)

The heating system switches on when the temperature falls below

18°\text{C}

. Determine the first time after midnight when the heating switches on. Give your answer in hours and minutes, correct to the nearest minute. [2 marks]

Solutions

30MasterySAQ-SSine, cosine and tangent functions6 marksPaper 2~9 min

The voltage

V

(in volts) in an electrical circuit is given by

V(t) = 240\sin(100\pi t)

, where

t

is the time in seconds.

(a)

State the amplitude and period of

V(t)

. [2 marks]

(b)

Calculate the voltage at

t = 0.003

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

(c)

Determine the first positive value of

t

for which the voltage equals

120

volts. Give your answer correct to 4 significant figures. [2 marks]

Solutions

Functions — Free Maths AI SL Practice Questions