Higher Level Mathematics

Ibrahim Wazir, Tim Garry, Peter Ashbourne

Chapter 2

Functions - all with Video Answers

Educators

Section 1

Definition of a function

Problem 1

For each equation $1-9,$ a) match it with its graph (choices are labelled $A$ to $L$ ), and
b) state whether or not the equation represents a function - with a justification. Assume that $x$ is the independent variable and $y$ is the dependent variable.
$$y=2 x$$
(graph cannot copy)

Willis James

Willis James

Numerade Educator

Problem 2

Match it with its graph (choices are labelled $A$ to $L$ ), and b) state whether or not the equation represents a function - with a justification. Assume that $x$ is the independent variable and $y$ is the dependent variable.
$$y=-3$$
(graph cannot copy)

Dale Sanford

Numerade Educator

Problem 3

Match it with its graph (choices are labelled $A$ to $L$ ), and b) state whether or not the equation represents a function - with a justification. Assume that $x$ is the independent variable and $y$ is the dependent variable.
$$x-y=2$$
(graph cannot copy)

Uma Kumari

Numerade Educator

Problem 4

Match it with its graph (choices are labelled $A$ to $L$ ), and b) state whether or not the equation represents a function - with a justification. Assume that $x$ is the independent variable and $y$ is the dependent variable.
$$x^{2}+y^{2}=4$$
(graph cannot copy)

Lucas Finney

Numerade Educator

Problem 5

Match it with its graph (choices are labelled $A$ to $L$ ), and b) state whether or not the equation represents a function - with a justification. Assume that $x$ is the independent variable and $y$ is the dependent variable.
$$y=2-x$$
(graph cannot copy)

Uma Kumari

Numerade Educator

Problem 6

Match it with its graph (choices are labelled $A$ to $L$ ), and b) state whether or not the equation represents a function - with a justification. Assume that $x$ is the independent variable and $y$ is the dependent variable.
$$y=x^{2}+2$$
(graph cannot copy)

Uma Kumari

Numerade Educator

Problem 7

Match it with its graph (choices are labelled $A$ to $L$ ), and b) state whether or not the equation represents a function - with a justification. Assume that $x$ is the independent variable and $y$ is the dependent variable.
$$y^{3}=x$$
(graph cannot copy)

Gregory Higby

Numerade Educator

Problem 8

Match it with its graph (choices are labelled $A$ to $L$ ), and b) state whether or not the equation represents a function - with a justification. Assume that $x$ is the independent variable and $y$ is the dependent variable.
$$y=\frac{2}{x}$$
(graph cannot copy)

Uma Kumari

Numerade Educator

Problem 9

Match it with its graph (choices are labelled $A$ to $L$ ), and b) state whether or not the equation represents a function - with a justification. Assume that $x$ is the independent variable and $y$ is the dependent variable.
$$x^{2}+y=2$$
(graph cannot copy)

Uma Kumari

Numerade Educator

Problem 10

Express the area, $A$, of a circle as a function of its circumference, $C$.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 11

Express the area, $A$, of an equilateral triangle as a function of the length, $\ell$, of each of its sides.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 12

A rectangular swimming pool with dimensions 12 metres by 18 metres is surrounded by a pavement of uniform width $x$ metres. Find the area of the pavement, $A,$ as a function of $x$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 13

In a right isosceles triangle, the two equal sides have length $x$ units and the hypotenuse has length $h$ units. Write $h$ as a function of $x .$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 14

The pressure $P$ (measured in kilopascals, kPa) for a particular sample of gas is directly proportional to the temperature $T$ (measured in kelvin, $\mathrm{K}$ ) and inversely proportional to the volume $V$ (measured in litres, $\ell$ ). With k representing the constant of proportionality, this relationship can be written in the form of the equation $P=k \frac{T}{V}$
a) Find the constant of proportionality, $k$, if $150 \ell$ of gas exerts a pressure of
$23.5 \mathrm{kPa}$ at a temperature of $375 \mathrm{K}$
b) Using the value of $k$ from part a) and assuming that the temperature is held constant at $375 \mathrm{K}$, write the volume $V$ as a function of pressure $P$ for this sample of gas.

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Bcrypt_Sha256$$2B$12$Koudzt7Vugfesdqzt.Btsohdsno3/5Wc5Bsgjhyqjgxswzij15Z06 Bcrypt_Sha256$$2B$12$Koudzt7Vugfesdqzt.Btsoec1F7Nikxndin/Owbntbjji9Jzcznki

Numerade Educator

Problem 15

In physics, Hooke's law states that the force $F$ (measured in newtons, N) needed to keep a spring stretched a displacement of $x$ units beyond its natural length is directly proportional to the displacement $x$. Label the constant of proportionality
$k$ (known as the spring constant for a particular spring).
a) Write $F$ as a function of $x$
b) If a spring has a natural length of $12 \mathrm{cm}$ and a force of $25 \mathrm{N}$ is needed to keep the spring stretched to a length of $16 \mathrm{cm},$ find the spring constant $k$
c) What force is needed to keep the spring stretched to a length of $18 \mathrm{cm} ?$

Tommy Erdos

Numerade Educator

Problem 16

Find the domain of the function.
$$\{(-6.2,-7),(-1.5,-2),(0.7,0),(3.2,3),(3.8,3)\}$$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 17

Find the domain of the function.
Surface area of a sphere: $S=4 \pi r^{2}$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 18

Find the domain of the function.
$$f(x)=\frac{2}{5} x-7$$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 19

Find the domain of the function.
$$h: x \mapsto x^{2}-4$$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 20

Find the domain of the function.
$$g(t)=\sqrt{3-t}$$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 21

Find the domain of the function.
$$h(t)=\sqrt[3]{t}$$

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Ankit Gupta

Numerade Educator

Problem 22

Find the domain of the function.
$$f: x \mapsto \frac{6}{x^{2}-9}$$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 23

Find the domain of the function.
$$f(x)=\sqrt{\frac{1}{x^{2}}-1}$$

Ahmad Reda

Numerade Educator

Problem 24

Do all linear equations represent a function? Explain.

Jeffrey Morris

Numerade Educator

Problem 25

Consider the function $h(x)=\sqrt{x-4}$
a) Find: (i) $h(21)$
(ii) $h(53)$
(iii) $h(4)$
b) Find the values of $x$ for which $h$ is undefined.
c) State the domain and range of $h$

James Kiss

Numerade Educator

Problem 26

Find the domain and range of the function, and b) sketch a comprehensive graph of the function clearly indicating any intercepts or asymptotes.
$$f: x \mapsto \frac{1}{x-5}$$

Fasiha Binat Zafar

Fasiha Binat Zafar

Numerade Educator

Problem 27

Find the domain and range of the function, and b) sketch a comprehensive graph of the function clearly indicating any intercepts or asymptotes.
$$g(x)=\frac{1}{\sqrt{x^{2}-9}}$$

James Kiss

Numerade Educator

Problem 28

Find the domain and range of the function, and b) sketch a comprehensive graph of the function clearly indicating any intercepts or asymptotes.
$$h(x)=\frac{2 x-1}{x+2}$$

James Kiss

Numerade Educator

Problem 29

Find the domain and range of the function, and b) sketch a comprehensive graph of the function clearly indicating any intercepts or asymptotes.
$$p: x \mapsto \sqrt{5-2 x^{2}}$$

Amy Jiang

Numerade Educator

Problem 30

Find the domain and range of the function, and b) sketch a comprehensive graph of the function clearly indicating any intercepts or asymptotes.
$$f(x)=\frac{1}{x}-4$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator