Trinity College Foundation Studies

Mathematics 1
Practice Booklet

©Trinity College

Welcome to Mathematics 1. This book contains a collection of exercises that will be used throughout the year to reinforce your understanding of the topics and
procedures we will be studying. The exercises are divided into sections:
• Pre-tutorial Questions
These are exercises that you should be doing between your lecture and the tutorial for a topic in order to be properly prepared for that week. This section contains
a mix of basic concept questions and ’warm-up’ questions. If you are struggling with the pre-tutorial questions you should contact your tutor for help before your
tutorial for that week.

• Examinable Section
This contains the majority of the exercises for the booklet and covers questions up to and including the standard of what we would consider examinable for this
subject. You should aim to complete the questions in this section each week. If you are comfortable with all of the examinable section questions for a topic you
consider yourself to be progressing appropriately.
• Non-examinable Section
This section contains questions that we would generally consider beyond the scope of what we would include on an exam. They may be more difficult than an
examinable question, require attention to the subtleties of the topic, or simply take more time than we would consider reasonable for an exam. They are included to
highlight particular issues with the topic or to provide more ’interesting’ or topical examples than the previous section. You should not focus your attention on this
section of exercises until after you have completed the Examinable section of questions.
If you ever have questions about anything in this booklet, contact your tutor. We hope you enjoy your time in Maths 1.

The Mathematics 1 Team

1
 4. Is the set {∅} empty? Explain.
5. If A ⊆ B and B ⊆ A, what conclusion can you draw?
Practice Book 6. For the sets A = {1, 2} and B = {2, 3}, select which of the following
statements are correct:
Semester 1 Part 1 a) 2 ∈ A b) 2 ⊆ A
c) 2 = A ∩ B d) {2} ∈ A
Chapter 1 e) {2} ⊆ A f) {2} = A ∩ B

7. Determine the value(s) of x ∈ Z, if any, for which {2x, (2x)2 } contains two
elements.
Pre-tutorial Questions
8. Decide whether the following statements are true. For any that are false,
There are no pre-tutorial questions for this chapter. provide an example where the statement in question does not hold.
a) A ∩ (B ∪ C) = (A ∩ B) ∪ C
Examinable Section b) A ∩ (B ∩ C) = (A ∩ B) ∩ C
c) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
1. How would you read the following in English? Explain the meaning for
each. 9. The set A = {1, 2} has four subsets. List them.

a) x ∈ A b) A ⊆ B 10. Use a number line to represent the following intervals A, B, C, and D:

c) A ∩ B d) A ∪ B A = [0, 3), B = (−3, ∞), C = (−∞, 1], and D = [−2, 2].
e) A \ B f) ∅
Hence find the following:
2. Let A = {1, 2, 3, 4, 5}. For each of the sets below, calculate A ∩ B, A ∪ B a) A ∩ B b) A ∩ C
and A \ B.
c) A ∩ D d) B ∩ C
a) B = {0, 2, 4, 6, 8} e) B ∩ D f) C ∩ D
b) B = {1, 3, 5} g) A ∩ B ∩ C ∩ D h) A ∪ B
c) B = {6, 7, 8, 9, 10} i) A ∪ C j) A ∪ D
k) B ∪ C l) B ∪ D
3. State one element of each of the following sets.
m) C ∪ D n) A ∪ B ∪ C ∪ D
a) Z \ N o) A \ B p) B \ A
b) Q \ Z q) A \ C r) C \ A
c) R \ Q s) A \ D t) D \ A

2
11. State the domain, codomain and range for each of the following functions: y

a) f : R → R, f (x) = 2x − 1
b) f : [0, ∞) → R, f (x) = 2x − 1 2
+
c) f : R → R, f (x) = 2x − 1
d) f : (−∞, 0] → R, f (x) = x2 − 1
e) f : R− → R, f (x) = 2x − 1
f) f : [−3, 2) → R, f (x) = 2x − 1
x
g) f : [−3, 2) → R, f (x) = x2 − 1 −2 2
√
h) f : R \ {0} → R, f (x) = 2x − 1 y = 4 − x2

12. Consider the function

f : [−4, 3] → R where f (x) = x2 . a) State the domain of h.

b) State the range of h.
a) State the domain of f .
b) Find f (−4) 15. Consider the graph of y = x3 − 12x + 16 shown here:
c) Find f (3) y
d) Sketch the graph of y = f (x).
e) State the range of f . 32

13. Consider the function

16
g : [−2, 2] → R where g(x) = −x2 .

(a) State the domain of g. x

−4 −2 2
(b) Find g(−2) 3
y = x − 12x + 16
(c) Find g(2)
(d) Sketch the graph of y = g(x).
(e) State the range of g.
State the domain and range of the following functions:
14. Consider the function a) f : R → R given by
p f (x) = x3 − 12x + 16.
h : [−2, 2] → R where h(x) = 4 − x2
b) g : [−4, ∞) → R given by
whose graph is shown here: g(x) = x3 − 12x + 16.

3
 c) h : [−3, 3] → R given by
h(x) = x3 − 12x + 16.
d) p : [−3, 5] → R given by
Chapter 2
p(x) = x3 − 12x + 16.

Pre-tutorial Questions
Non-examinable Section
Tutorial 2
Investigation Questions
17. Solve the following quadratic equations for x:
16. The power set of a set S is defined as the set of all subsets of S. It is
denoted P(S). (a) x2 + 4x + 3 = 0
(b) x2 + 13x + 42 = 0
a) Write the power set of B = {1, 2, 3}. How many elements does it
have? 18. Factorise the following expressions:
b) If the set C has n elements, how many elements does P(C) have?
(a) x2 + 4x + 3
Justify your answer.
(b) x2 + 13x + 42
(c) x3 + 4x2 + 3x
(d) x3 − 2x2 − x + 2

19. Solve for x:

(a) 2x3 − 4x2 + 2 = 0
(b) x4 − 10x2 + 9 = 0

20. Solve for x:

√
(a) x = x
√
(b) x = − x

21. Complete the square for the following quadratic expressions:

(a) x2 + 4x + 7
(b) 3x2 + 6x + 4

22. Factorise:
(a) t3 + 1

4
 (b) x2 − 4y 2 (f) x2 + x + 1 = 0
(g) x2 + 2x + 1 = 0
23. Find the expanded form of each of the following expressions:
(h) x2 − 2x − 2 = 0
6
(a) (a + x)
(b) (a − x)6 28. By writing each of the following equations in the form ax2 + bx + c = 0,
find the discriminant b2 − 4ac. Thus determine the nature of the roots.
Tutorial 3
(a) 2x2 − 7x = −4
24. Solve the following inequalities for x: (b) 3x − x2 = 4
25
(a) 1 + x < 7x + 5 (c) 3x + = 10
3x
(b) 4 ≤ 3x − 2 < 13
29. Factorise these expressions over R:
(c) x2 + 3x < 4
(a) x2 + 2x + 1
25. Solve the following inequalities for x:
(b) x2 − 2x − 2
1+x (c) x2 − 3
(a) >1
1−x
x (d) t2 − 6t + 2
(b) <4 (e) 12x2 + 5x − 3
3+x
(f) 3x2 y 4 + 6x3 y 3
26. Solve for x:
(g) x2 − 1
(a) |3x − 6| = 6
(b) |x − 1| ≤ 3 30. Complete the square for the following quadratics:
(c) |6x + 1| > 7 (a) x2 − 4x + 7
(b) x2 − 6x + 10
Examinable Section (c) x2 + 4x − 3
(d) 2x2 + 8x − 6
27. Solve the following equations for x
(e) 3x2 + 3x + 1
2
(a) 6x − 5x − 6 = 0
(b) 2x2 + 3x + 1 = 0 31. Solve for x:
√
(c) x2 + 9x − 10 = 0 (a) 2x − 2 = x − 1
√
(d) x2 + 9x − 1 = 0 (b) 4x + 1 = 3 − 3x
√
(e) 2x2 − 2x − 2 = 0 (c) x + 1 = 1 − x

5
 √
(d) 3x − 5 = x − 1 (h) x4 − 2x2 − 3x − 2
√ √
(e) 2x + 1 − x = 1 (i) t3 + 3t2 + 3t + 126
√
(f) 3x − 2 = −x (j) x4 − 1
(k) x3 − 6x2 + 12x − 8
32. Solve the following equations for their unknowns:
√ (l) x3 y − y 3 x
(a) 4k + 5 − 12 k = 2
√ 36. Use Pascal’s Triangle to find the expanded form of each of the following
(b) 1 + 2 − 3x = x.
expressions:
33. Use long division to divide (a) (a + x)3
(a) x3 + 2x2 + 2x + 2 by x + 1 (b) (a + x)4
(b) 2x3 + 9x2 + 10x + 23 by x + 4 (c) (1 + x)3
(c) x5 + 3x2 + 2x + 1 by x2 + 1 (d) (5 − 2m)6
(e) ( 14 + 3x2 )5
34. Solve for x:
1 4
(f) (3t − 5t ) .
3 2
(a) x + 3x + 3x + 2 = 0 (g) (1 + x )2 4

(b) x3 + 4x2 + 5x + 6 = 0 (h) (2 − x)4

4 2
(c) x − 2x − 3x − 2 = 0
37. Solve the following inequalities for x:
(d) 2x3 − 9x2 + 9x − 2 = 0
(e) 2x3 − 7x2 − 3x + 18 = 0 (a) 2x + 1 ≤ 4x − 3 ≤ x + 7
4 3 2
(f) x + 10x + 35x + 50x + 24 = 0 (b) x2 + 5x > −6,
(g) x4 + 4x3 + 6x2 + 4x + 1 = 0 38. Solve the following inequations for x. Write your answers using the
bracket notation for intervals.
35. Factorise these expressions over R:
(a) (x + 3)(3x − 2)(1 − x) ≥ 0
(a) 2m3 − 6m2 + 2m + 2.
(b) (x − 1)(x − 2)(3 + 2x) ≤ 0
(b) 2w3 + 30w2 + 50w − 474
(c) (x2 + x + 1)(x − 1)(x + 2) < 0
(c) 2x3 − 4x2 − 146x − 140
(d) (x2 − 4)(2x + 1) < 0
(d) p3 − 13p2 + 49p − 54
(e) (x3 − 1)(x2 − 9) ≥ 0
(e) x3 − 27
(f) x3 + 2x2 − 5x − 6
(g) x3 + 2x2 − x − 2

6
39. Solve the following inequations for x. Write your answers using the (c) | − 4x − 5| < |1 − 3x|
bracket notation for intervals. (d) |4x + 5| ≥ | − 1 + 3x|
3 2
(a) 2x − 9x + 13x − 6 ≥ 0 4x + 3
(e) ≥1
3 2
(b) −2x − x + 5x − 2 < 0 −2 − 5x
(c) 12x3 + 4x2 − 9x − 3 > 0 5 1
(f) >
−2 + x 1+x
(d) −2x3 + 12 ≤ −8x + 3x2
(e) x4 + 3x3 + 3x2 − x − 6 > 0
Non-examinable Section
40. Solve the following inequations for x. Write your answers using the
bracket notation for intervals. 43. Show that all of the roots of x3 + 2x2 − 5x − 3 = 0 are irrational.
2 44. Fully factorise each expression over R:
(a) ≥2
2x + 3
x+1 (a) x4 − 7x2 + 6
(b) <3
3x − 5 (b) x4 − x2 + 9.
x2 + x − 6 (c) x4 + 64
(c) ≤0
4 + 3x (d) 9x4 + 2x2 + 1
3x − 1 x + 1 1 − 2x
(d) − ≥ (e) x8 − 1.
2 4 3
2x + 3 3x + 1 (f) x4 + x2 + 4
(e) − ≤ −3
2 3x (g) 36x4 + 15x2 + 4
41. Solve for x: (h) t4 − t2 + 1

(a) |3x + 7| = 4 (i) t6 + 1

(b) | − 2x − 3| = 2 45. Consider the equation

(c) |2x + 3| = x + 2
20x3 + 193x2 + 290x = 126.
(d) | − 2x + 5| = 3 − x
x+1 Use a calculator or other technology to find the roots, and write down
(e) =3 each of them to 2 decimal places. Then use algebra to solve the equation
2x + 1
for x, giving the exact value of each root.
42. Solve the following inequations for x:
46. Determine the values of x ∈ R that satisfy the following inequations. Plot
(a) |7x + 2| ≥ 4 these solutions on separate number lines.
4
(b) |3x + 5| ≤ (a) −1 < x2 + 2x + 1 < 2
7

7
 (b) x2 + 3x + 2 < 0
(c) x(x − 4) > 5
Chapter 3
(d) 0 < x2 − 7x + 10 < 1
(e) −1 < x2 + 9x + 19 < 0
(f) −3 < x2 + 7x + 9 < 9 Pre-tutorial Questions

47. Solve 3x2 − 2x − 1 ≤ 0 on Z (the set of integers). 49. Convert the following angles from degrees to radians:
a) 45◦
48. Solve the following inequalities for x and write your answers in interval
notation: b) 60◦
c) 150◦
(a) |x2 − 2| − x < 0
(b) |x2 − 6x + 6| < 2 50. Convert the following angles from radians to degrees:
3π
a) 2 rad
2π
b) 3 rad

51. Using your calculator write the following o four decimal places:
a) sin 53◦
b) cot 0.4

52. Without using a calculator, find the exact values of the expressions given
below:
a) cosec π2

b) sin π
c) tan π

53. For each of the angles θ considered below, find

sin θ, cos θ, tan θ, cosec θ, sec θ and cot θ.
a) The angle θ between the x-axis and the ray extending from the ori-
√
gin to the point (1, 3).
b) The angle θ between the x-axis and the ray extending from the ori-
√
gin to the point ( 3, 1).

8
Examinable Section a) Find the depth when t = 0.
b) Find the depth at 1 am. Write your answer to 2 decimal places.
54. Convert the following angles from degrees to radians:
c) Find the depth at 2 am.
a) 30◦ b) 90◦ d) Find the maximum depth.
c) 120◦ d) 135◦ e) Find the minimum depth.
e) 270◦ f) 360◦
59. Express each of the following in terms of sin θ, cos θ or tan θ
55. Convert the following angles from radians to degrees:
a) sin(2π − θ) b) cos(2π − θ)
π π π  π 
a) radians b) radians c) sin −θ d) cos −θ
4 3
5π  2π  2
c) radians d) π radians e) sec −θ f) tan(π − θ)
6
 2π  π 
56. Using your calculator find the value (to 4 decimal places) of the following: g) cos +θ h) sin +θ
2 2
a) cos 4.86 b) sin 5.78 60. Without using a calculator, find the exact values of the expressions given
◦
c) tan 49 d) cot 3.64 below:
   
e) cos 5.316 2π 5π
a) sin b) cot
3 6
57. A body moves so that its speed, V m · s−1 , after t seconds is given by    
5π 7π
  c) tan d) sin
t 4 6
V = 30 − 8 cos .    
2 7π 3π
e) cos f) sec
6 4
a) Calculate the body’s speed after four seconds. Present your answer    
in a sentence and to two decimal places. 5π 4π
g) sec h) cot
4 3
b) Calculate the body’s initial speed. (That is, its speed when t = 0.)
c) Calculate the body’s greatest speed. (Do not use calculus to answer 61. For each of the following angles θ, find exact values of sin θ, cos θ and tan θ
this question.) without the use of a calculator.
3π 5π
58. The depth of water in a particular part of a bay varies with time according a) b)
4 6
to the following formula: 7π 5π
c) d)
π  6 4
D = 20 + 3 sin t 4π 5π
12 e) f)
3 3
where D is the depth measured in metres, and t is the number of hours 7π 11π
after midnight, with 0 ≤ t ≤ 24. g) h)
4 6

9
62. Without using a calculator, find the exact values of the expressions given 66. Solve the following for x ∈ [0, 2π]:
below:
1
a) sin x = −
   
7π 11π
a) cot b) cos 2
4 6
    b) sin(3x) = 0
5π 5π
c) sin d) cosec √  π
3 3 c) 3 tan 3x − = −1
  6
13π
e) sin (2π) f) sin
6  π h π πi
67. Solve tan 2θ + = 1 for θ ∈ − , .
4 2 2
   
9π 7π
g) tan h) cot
4 3
68. Given that
 π  √5 − 1
   
13π 5π
i) cos j) cos
6 2 sin =
10 4
1
63. a) Solve cos x = for 0 ≤ x ≤ 2π. find the exact value of cos π5 .


2
1
b) Solve cos x = 2 for 0 ≤ x ≤ 3π.
1 69. Prove the following identities:
c) Solve cos x = 2 for −2π ≤ x ≤ 3π.
sin(3A) cos(3A)
64. Solve a) − =2
 π 1 h π πi sin A cos A
sin 7x − = for x ∈ − ,
6 2 2 2 b) cos(4x) = 8 cos4 x − 8 cos2 x + 1
using the following steps: 1 + cos t
c) (cot t + cosec t)2 =
a) Rewrite the equation in terms of a new unknown A = 7x − π 1 − cos t
6.
b) Solve the new equation for A, writing the values in a set from least 1 1
d) + = 2 sec2 θ
to greatest. 1 − sin θ 1 + sin θ
c) Hence complete the original task. e) (1 − tan θ)2 + (1 + tan θ)2 = 2 sec2 θ
f) sin(α + β) + sin(α − β) = 2 sin α cos β
65. Solve the following equations for θ ∈ [0, 2π]:
1 70. a) Using the same set of axes, sketch the graphs of y = sin x and y =
a) sin (2θ) = −
2 cos x between x = 0 and x = 2π.
 π
b) tan 3θ + = −1
4 b) Find the x–coordinates of the points of intersection of the two
√ graphs.
c) sin θ + 3 cos θ = 0
c) Hence find the values of x between 0 and 2π for which sin x > cos x.

10
Non-examinable Section 77. Solve each of the following inequations:
 π √
71. Solve each of the following inequalities, for x ∈ [0, 2π]: a) 2 cos 2x − > − 3 for x ∈ [0, 2π]
3
1
a) cos x − √ ≥ 0
 π
2 b) 2 sin 3t + ≤ 1 for t, where 0 ≤ t ≤ 2π
4
1 x π
b) sin x − √ ≥ 0 c) tan − ≤ 1 for x, where 0 ≤ x ≤ 2π
2 2 4
√  
72. a) Solve tan(2x) − 3 ≤ 0 for x ∈ [0, 2π]. πt π
√ d) 2 sin − + 4 > 3 for t, where 12 < t ≤ 40
6 3
b) Solve 2 sin(2x − π) + 1 > 0 for x ∈ [0, 2π].
 
θ π
73. Solve the following inequalities, for θ ∈ [0, 2π]: e) cos − < 1 for θ ∈ [−6π, 2π].
4 6
 π 1
a) sin θ − − <0
6 2 x  
π 1 8π 16π
√  π 78. Solve sin − > 2 for x ∈ − , .
b) 2 cos θ − +1≥0 4 3 3 3
4
 π 
c) tan 2 θ + ≤1 1
2 79. Solve cos4 θ − sin4 θ < for θ ∈ [−π, π].
2
74. Suppose that A is the angle in −π, − π2 for which cosec A = −3.


 π π
80. Find, to two decimal places, all real values of θ in the interval − ,
a) Draw a diagram that shows the angle A. 2 2
for which tan(θ) + tan(2θ) = 2 by substituting x for tan(θ) and then using
b) Evaluate cos A and cot A.
a calculator or other technology to solve the resulting cubic equation.
 
75. 5π
a) Determine the exact value of tan . π
12 81. Determine the exact value of 1 − sin .
π   16
3π
b) Let x be sin . Express sin in terms of x.
7 7 82. One way to prove the two double angle formulas for 2θ ∈ (0, π2 ) is by
c) Prove the following identity: using the following two properties of circles: Suppose a line segment AB
forms the diameter of a circle with center O, and let C be a point on the
cos6 t + sin6 t = 1 − 3 cos2 t sin2 t. edge of the circle. Then:

• the angle ∠COB is twice that of the angle ∠CAB.

76. With θ as the variable on the horizontal axis, sketch the graph of y = • the angle ∠ACB is a right angle.
tan(θ) from θ =√−2π to θ = 2π. On your graph, highlight the points for
which tan(θ) < 3. Consider the following picture:

11
 C 83. One geometric proof of the addition formulae involves setting up the fol-
lowing rectangle:
α
x1 x2

θ 2θ
A O B

y2

y π
1 2

Using these facts above, and the definitions of sin and cos. y1

a) Find the value of α

β
α
b) Identify the length corresponding to sin(2θ) and calculate it in terms
of cos θ and sin(θ). x

c) Identify the length corresponding to cos(2θ) and calculate it in terms Calculate each of the unknown lengths in terms of sin α, cos α, sin β and
of cos θ and sin(θ). cos β. Then compare the lengths of the opposite sides of the rectangle.

(Hint: Make your life easier by making the circle in your picture a unit
circle.)

12
 a) log10 2 + log10 3 − log10 6
b) 4 log10 3 − log10 81
Chapter 4
1 1
c) log3 − log7 49 + log5 √
9 5
Pre-tutorial Questions d)
log10 8
log10 4
84. Simplify the following expressions. 5 √
e) log10 a − log10 a where a ∈ (0, ∞)
a) x5 y × x2 y 4 2
b) (6x)2 × (2x2 )3 f) 3log3 (5a−2) where a ∈ ( 25 , ∞)

c) (−a3 )5 × 2(a−3 )0 89. Simplify the following.

 m 2  2n −2
log2 9
d) (a)
2n m log2 81
2 log3 8
85. Simplify the following expressions. (b)
log3 16
a) 24n × 43n × 8−n
63n × 9n+2 × 8 Examinable Section
b)
81n × 4n+1
90. Write the following as sums and/or differences of logarithms.
86. Evaluate the following without using your calculator.
a) log3 (x4 (x + 3)) where x ∈ (0, ∞)
5
a) 2  3 
x
b) 9− 2
1 b) log8 where x ∈ (8, ∞)
x−8
2 √ 
c) 32 5 a
c) ln where a, b, c ∈ (0, ∞)
3 b2 c5
d) 25− 2
2
e) (−64)− 3 91. Use exponential and logarithmic laws, and the fact that aloga (x) = x
(where a ∈ R+ \ {1}, x > 0), to simplify the following expressions.
87. Simplify the following expressions.
√ √ a) eln(2x) where x ∈ (0, ∞)
5 3
a) m2 × m5 b) ln(e3y−1 )
√3
√
5
b) 27a2 b0 ÷ 32ab3 c) ln(e2 e5 )

88. Simplify the following to a single logarithm and evaluate where possible d) ep ln(m) where m ∈ (0, ∞)
without using a calculator. e) eln x+ln 5 where x ∈ (0, ∞)

13
92. Solve the following exponential equations for x ∈ R. a) 16x < 2
1
a) 2x = 0.125 b) 0.54x−1 <
16
b) 4x = 83−x × 2x−1 1
c) 42x+1 ≥
c) 9 2x+1
= 27x 32

3x−1 98. Sketch the graph of y = log2 x. Remember to label the x–intercept and
d) =1 the asymptote. Label two more points on the curve: (8, 3) and (x, y) for a
272x
general x greater than 8. Hence write down the solution to log2 x > 3.
93. Solve the following equations for x ∈ R.
(Hint: Use substitution) 99. Sketch the graph of y = log4 x. Label two further points on the curve:
(16, 2) and (x, y) for a general positive x less than 16. Hence state the
a) 22x = 2x+1 − 1 solution to log4 x < 2.
b) 4x − 9(2x ) + 8 = 0
100. Solve for x ∈ R.
94. Solve each of the following for x ∈ R. a) log5 (x + 3) ≤ 2
a) ln(x2 − 3x − 9) = 0 b) 5 + 3 log0.5 x ≥ 11

b) log3 (5x + 11) − log3 x = 3

Non-examinable Section
95. Solve for x ∈ R, giving your answers correct to 2 decimal places.
Challenge
a) 5x = 12
b) 3−2x = 6 101. (a) Show that for all values of the parameter c, the equation
!x
96. Sketch the graph of y = 2 x . Remember to show the asymptote and to x
1
2 − =c
include a label for the intercept(s). 2
Then solve the following inequations for x ∈ R.
has one solution for the unknown x.
x
a) 2 ≤ 8 (b) Suppose that 2x +2−x = c. For each value of c, determine the number
x
b) 2 ≥ 0.25 of solutions for x.
c) 2x > −3 102. Solve the following inequalities for x ∈ R.
x
d) 2 < −3
a) ln(x + 3) + ln(x + 5) > 2
e) 2x < 1 b) log4 (1 + x) − log4 (1 − x) ≤ 0

97. Solve for x ∈ R in each of the following.

14
 1
(a) f (x) =
x
Chapter 5 (b) f (x) =
x
2x − 1
(c) f (x) = 2x − 1
Pre-tutorial Questions (d) f (x) = sin x
√
There are no pre-tutorial questions for this chapter. (e) f (x) = x
√
(f) f (x) = −x
√
Examinable Section (g) f (x) = x2 − 1
(h) f (x) = log4 (2x + 1)
103. Are the following functions one–one?
1
(a) f : [0, 2π] −→ R where f (x) = sin x (i) f (x) = √
x
(b) f : [0, π] −→ R where f (x) = sin x x
(j) f (x) = √
(c) f : [0, 2π] −→ R where f (x) = cos x 1−x
(d) f : [0, π] −→ R where f (x) = cos x 1
(k) f (x) = 2
x − 4x + 3
104. Find the implied domain for the following functions:
f
4x + 1 107. In each of the following, find the rules for f + g, f −g, f g and , and state
(a) f (x) = g
x+4 the corresponding domains:
√
(b) f (x) = 2 − x
x 1
4 (a) f (x) = and g(x) =
(c) f (x) = x−4 2x + 3
1 − x2
√ 1
(d) f (x) = 1 − x2 (b) f (x) = log2 (x + 2) and g(x) =
4x x2 − 1
(e) f (x) = √ √
1 − x2 (c) f (x) = x2 − 2x + 1 and g(x) = x
(f) f (x) = 1 + x2
√
105. Find the implied domain of the following functions: (d) f (x) = 9 − x2 and g(x) = 2x

(a) f (x) = log2 (2x − 1) √

(e) f (x) = log3 x and g(x) = 1−x
(b) f (x) = log10 (1 − x)
(c) f (x) = 2 log4 (x − 1) 108. Find the implied domain of the function
p
106. Find the implied domain of the following functions: f (x) = x3 − 19x + 30.

15
109. If f (x) = x2 and g(x) = 2x + 4 then 114. The functions f and g below have been expressed in a particular format.

(a) find the rule for f (g(x)). f : (−5, 10] → R, f (x) = 2x + 3

(b) find the rule for g(f (x)). g : [4, 28] → R, g(x) = 3x − 7
√
110. Consider the function f (x) = 1 + x + 1. Use that same format to express the following composite functions.

(a) Find dom(f ). a) g ◦ f b) f ◦ g

(b) Find ran(f ). c) g ◦ g d) f ◦ f
(c) Find dom(f −1 ). 115. This exercise is about the function
−1
(d) Find ran(f ). 2x − 3
−1 f (x) =
(e) Find the rule for f . 5x − 9
(f) Sketch the graphs of y = f (x) and y = f −1 (x). that has maximal domain. Find the domain of f, the rule for f −1 (x), and
the range of f −1 .
111. Consider the functions
√ 116. Which of the following functions has an inverse function?
f : (4, ∞) −→ R where f (x) = x
a) f : R → R where f (x) = x2 − 2
and b) f : [0, ∞) → R where f (x) = x2 + 1
g : R− −→ R where g(x) = x2 .
c) f : R → R where f (x) = x3
(a) Find the domain of f ◦ g. d) f : [−1, ∞) → R where f (x) = (x + 3)2
(b) Find the rule for f (g(x)).
117. Consider the function given by
112. Consider the functions p
√ f (x) = 4 − x2 .
f : [1, ∞) −→ R where f (x) = x
a) Sketch the graph of y = f (x).
and b) Find the domain and range of f .
g : R −→ R where g(x) = x2 .
c) Does f have an inverse function? Justify your answer.
(a) Find the rule for f ◦ g.
118. (a) Find the smallest number b such that the function f : [b, ∞) → R
(b) Find dom(f ◦ g). given by f (x) = x2 − 4 has an inverse function. Find the rule for the
 2 inverse function.
x − 2 if x > 1
113. If f (x) = and g(x) = 2x (b) Find the largest number b such that the function f : (−∞, b] → R
1−x if x ≤ 1
given by f (x) = (x + 2)2 has an inverse function. Find the rule for
then find the rule for f (g(x)). the inverse function.

16
 1 Answers
119. (a) Let f : (−∞, −2) → R given by f (x) = .
(x + 2)2
i. Sketch the graphs of y = f (x) and y = f −1 (x). Chapter 1
ii. Completely determine f −1 . That is, find the domain, range and
rule for f −1 . Examinable Section
1
(b) Let g : (−2, ∞) → R given by g(x) = . 1. Find these terms in the course notes.
(x + 2)2
i. Sketch the graphs of y = g(x) and y = g −1 (x). 2. a) A ∩ B = {2, 4}
ii. Completely determine g −1 . That is, find the domain, range and A ∪ B = {0, 1, 2, 3, 4, 5, 6, 8}
rule for g −1 . A \ B = {1, 3, 5}

120. Find the rule for f −1 if f is the function defined by b) A ∩ B = {1, 3, 5}

A ∪ B = {1, 2, 3, 4, 5}
f : [−2, 0] → R where f (x) = 4 − x2 . A \ B = {2, 4}
c) A ∩ B = {}
121. Consider the function A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A\B =A
f : (−∞, b] → R where f (x) = x2 + 2x.

a) Find the largest value of b so that f has an inverse function. 3. Omitted

Using this value of b, 4. {∅} is not empty, it contains a single element: ∅.

b) state the domain and range of f . 5. If A ⊆ B and B ⊆ A, then A = B.

−1
c) state the domain and range of f .
−1
6. a), e) and f) are correct.
d) find f .
7. x ∈ Z \ {0}.
122. Consider the function
8. a) False. e.g. if A = {1, 2, 4, 5}, B = {2, 3, 5, 6}, C = {4, 5, 6, 7}
f : S → R where f (x) = 2x + 2.
b) True (This is called the associativity property)
If f has inverse function given by c) True (Compare this to the distributive law for multiplication and
addition)
1
f −1 : [0, ∞) → R where f −1 (x) = x − 1,
2
9. ∅, {1}, {2}, {1, 2}
then find the set S.
10. A number line representing the sets, A, B, and C is shown below.

17
 D• • d) y
• C
16
B
y = f (x)
• A 9
x
−4 −3 −2 −1 0 1 2 3 4

a) [0, 3) b) [0, 1] −4 3 x
c) [0, 2] d) (−3, 1] e) ran(f ) = [0, 16]
e) [−2, 2] f) [−2, 1]
g) [0, 1] h) (−3, ∞) 13. a) dom(g) = [−2, 2]
i) (−∞, 3) j) [−2, 3) b) −4
k) (−∞, ∞) = R l) (−3, ∞) c) −4
m) (−∞, 2] n) (−∞, ∞) = R d) y
o) ∅ p) (−3, 0) ∪ [3, ∞) −2 2
q) (1, 3) r) (−∞, 0) x
s) (2, 3) t) [−2, 0)

11. The codomain for all these functions is R.

−4
y = g(x)
a) dom f = R, ran f = R
b) dom f = [0, ∞), ran f = [−1, ∞) e) ran(g) = [−4, 0].
+
c) dom f = R , ran f = (−1, ∞)
d) dom f = (−∞, 0], ran f = [−1, ∞) 14. a) dom(h) = [−2, 2]
e) dom f = R− , ran f = (−∞, −1) b) ran(h) = [0, 2]
f) dom f = [−3, 2), ran f = [−7, 3)
15. a) dom(f ) = R and ran(f ) = R
g) dom f = [−3, 2), ran f = [−1, 8]
b) dom(g) = [−4, ∞) and ran(g) = [0, ∞)
h) dom f = R \ {0}, ran f = R \ {−1}
c) dom(h) = [−3, 3] and ran(h) = [0, 32]
12. a) dom(f ) = [−4, 3] d) dom(p) = [−3, 5] and ran(p) = [0, 81]
b) 16
c) 9 Non-examinable Section

18
16. P(B) has 8 elements: (c) x ∈ (−4, 1)
∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, B
25. (a) x ∈ (0, 1)
If B has n elements, P(B) has 2n elements.
(b) x ∈ (−∞, −4) ∪ (−3, ∞)
This is Cantor’s Theorem, and holds for infinite as well as finite sets (once
you have worked out what the ”number of elements” means for infinite 26. (a) x ∈ {4, 0}
sets). (b) x ∈ [−2, 4]
Chapter 2 4
(c) x ∈ (−∞, − ) ∪ (1, ∞)
3
Pre-tutorial Questions Examinable Section
17. (a) x = −1, −3
27. (a) x = 32 , − 32
(b) x = −6, −7
(b) x = − 12 , −1
18. (a) (x + 1)(x + 3) (c) x = 1, −10
√
(b) (x + 6)(x + 7) (d) x = −9± 85
2
√
(c) x(x + 1)(x + 3) 1± 5
(e) x = 2
(d) (x − 2)(x + 1)(x − 1)
(f) No real solutions.
√
1± 5
19. (a) x = 1, 2
(g) x = −1
√
(b) x = ±1, ±3 (h) x = 1 ± 3

20. (a) x = 0, 1 28. (a) 17, two distinct real roots

(b) x = 0 (b) −7, no real roots

21. (a) (x + 2)2 + 3 (c) 0, one real root

(b) 3[(x + 1)2 + 13 ] = 3(x + 1)2 + 1 29. (a) (x + 1)2

√ √
22. (a) (t + 1)(t2 − t + 1) (b) (x − 1 − 3)(x − 1 + 3)
√ √
(b) (x − 2y)(x + 2y) (c) (x − 3)(x + 3)
√ √
(d) (t − 3 − 7)(t − 3 + 7)
23. (a) a6 + 6a5 x + 15a4 x2 + 20a3 x3 + 15a2 x4 + 6ax5 + x6
(e) (3x − 1)(4x + 3)
(b) a6 − 6a5 x + 15a4 x2 − 20a3 x3 + 15a2 x4 − 6ax5 + x6
(f) 3x2 y 3 (y + 2x)
2
24. (a) x ∈ (− , ∞) (g) (x + 1)(x − 1)
3
(b) x ∈ [2, 5) 30. (a) (x − 2)2 + 3

19
 (b) (x − 3)2 + 1 (f) (x − 2)(x + 1)(x + 3)
2
(c) (x + 2) − 7 (g) (x − 1)(x + 1)(x + 2)
2
(d) 2(x + 2) − 14 (h) (x + 1)(x − 2)(x2 + x + 1)
(e) 3(x + 12 )2 + 1
4 (i) (t + 6)(t2 − 3t + 21)
31. (a) x = 1, 3 (j) (x2 + 1)(x + 1)(x − 1)
(b) x = 4 (k) (x − 2)3
9
(c) x = 0 (l) xy(x + y)(x − y)
(d) x = 2, 3 36. (a) a3 + 3a2 x + 3ax2 + x3
(e) x = 0, 4 (b) a4 + 4a3 x + 6a2 x2 + 4ax3 + x4
(f) No solutions. (c) 1 + 3x + 3x2 + x3
√ √
32. (a) k = −2 5 + 4, k = 2 5 + 4 (d) 64m6 − 960m5 + 6000m4 − 20000m3 + 37500m2 − 37500m + 15625
405 8 135 6 45 4 15 2 1
(b) There are no possible (real) values for x. (e) 243x10 + 4 x + 8 x + 32 x + 256 x + 1024

33. (a) x2 + x + 1 + 1 (f) 81t4 − 108 2 54 12 1

5 t + 25 − 125t2 + 625t4
x+1
2 4 6 8
(b) −1
2x2 + x + 6 + x+4 (g) 1 + 4x + 6x + 4x + x
(h) 16 − 32x + 24x2 − 8x3 + x4
(c) x3 − x + 3 + 3x−2
x2 +1
1
34. (a) x = −2 37. (a) x ∈ [2, 3 ]
3
(b) x = −3 (b) x ∈ (−∞, −3) ∪ (−2, ∞)
(c) x = −1, 2
√ 2
(d) x = 1, 7±4 33 38. (a) x ∈ (−∞, −3] ∪ [ , 1]
3
(e) x = 2, − 32 , 3 3
(b) x ∈ (−∞, − ] ∪ [1, 2]
(f) x = −1, −2, −3, −4 2
(c) x ∈ (−2, 1)
(g) x = −1
1
√ √ (d) x ∈ (−∞, −2) ∪ (− , 2)
35. (a) 2(m − 1)(m − 1 − 2)(m − 1 + 2) 2
√ √
(b) 2(w − 3)(w + 9 − 2)(w + 9 + 2) (e) x ∈ [−3, 1] ∪ [3, ∞)
(c) 2(x + 7)(x + 1)(x − 10) 3
√ √ 39. (a) x ∈ [1, ] ∪ [2, ∞)
(d) (p − 2)(p − 11
− 13
− 11 13 2
2 2 )(p 2 + 2 )
1
(e) (x − 3)(x2 + 3x + 9) (b) x ∈ (−2, ) ∪ (1, ∞)
2

20
 √ √
3 1 3 7 1
(c) x ∈ (− ,− ) ∪ ( , ∞) (f) x ∈ (−∞, − ) ∪ (− , 2) ∪ (2, ∞)
2 3 2 4 2
3 Non-examinable Section
(d) x ∈ [−2, − ] ∪ [2, ∞)
2
(e) x ∈ (−∞, −2) ∪ (1, ∞) 43. omitted
√ √
3 44. (a) (x + 1)(x − 1)(x + 6)(x − 6)
40. (a) x ∈ (− , −1] √ √
2 (b) (x2 + 7x + 3)(x2 − 7x + 3)
5 (c) (x2 − 4x + 8)(x2 + 4x + 8)
(b) x ∈ (−∞, ) ∪ (2, ∞)
3 (d) (3x2 + 2x + 1)(3x2 − 2x + 1)
4 √ √
(c) x ∈ (−∞, −3] ∪ (− , 2] (e) (x + 1)(x − 1)(x2 + 1)(x2 + 2x + 1)(x2 − 2x + 1)
3 √ √
13 (f) (x2 − 3x + 2)(x2 + 3x + 2)
(d) x ∈ [ , ∞)
23 (g) (6x2 − 3x + 2)(6x2 + 3x + 2)
√ √ √ √
−21 − 489 489 − 21 (h) (t2 − 3t + 1)(t2 + 3t + 1)
(e) x ∈ (−∞, ] ∪ (0, ] √ √
12 12 (i) (t2 + 1)(t2 − 3t + 1)(t2 + 3t + 1)
11
41. (a) x ∈ {− , −1} 45. (a) −7.65, −2.35, 0.35
3 √ √ 7
5 1 (b) x1 = −5 − 7, x2 = −5 + 7, x3 = 20 or x3 = 0.35
(b) x ∈ {− , − } √ √
2 2
46. (a) −1 − 2 < x < −1 + 2
5
(c) x ∈ {− , −1} (b) −2 < x < −1
3
8 (c) x > 5 or x < −1
(d) x ∈ {2, } √ √
3 (d) 5 < x < 7+ 13
or 7− 13
<x<2
2 2
4 2 √ √
(e) x ∈ {− , − } (e) −9− 5
< x < −5 or −4 < x < −9+ 5
7 5 2 2

6 2 (f) −7 < x < −4 or −3 < x < 0

42. (a) x ∈ (−∞, − ] ∪ [ , ∞)
7 7 47. If we solve the inequality over the reals we get − 13 ≤ x ≤ 1 the only
13 31 elements of Z in that interval are 0 and 1.
(b) x ∈ [− , − ]
7 21
4 48. (a) x ∈ (1, 2)
(c) x ∈ (−6, − ) √ √
7 (b) x ∈ (3 − 5, 2) ∪ (4, 3 + 5)
4
(d) x ∈ (−∞, −6] ∪ [− , ∞) Chapter 3
7
5 2 2 Pre-tutorial Questions
(e) x ∈ [− , − ) ∪ (− , 1]
9 5 5
49.
21
 a) π
58. a) When t = 0 the depth is 20 m.
4
b) π b) At 1 am the depth is ≈ 20.78 m.
3
c) 5π c) At 2 am the depth is 21.5 m.
6
d) The maximum depth is 23 m.
50. a) 270◦ e) The minimum depth is 17 m.
b) 120◦
59. a) − sin θ b) cos θ
51. a) 0.7986 c) cos θ d) sin θ
b) 2.3652 1
e) f) − tan θ
sin θ
52. a) 1 g) − sin θ h) cos θ
b) 0
√
3
√
60. a) 2 b) − 3
c) 0 c) 1 d) − 12
√
3
√
53. sin θ cos θ tan θ cosecθ sec θ cot θ e) − 2 f) − 2
√
3 1
√ √
(a) 3 √2 2 √1 g) − 2 h) √1
2 √2 3
1 3
√3 3
(b) √1 2 √2 3
2 2 3 3 61.
θ sin θ cos θ tan θ
Examinable Section 3π √1
(a) 4 2
− √12 −1
π π √
54. a) 6 b) 2 (b) 5π 1
− 3
− √13
6 2 √2
2π 3π
c) 3 d) 4 (c) 7π
− 12 − 23 √1
6 3
5π
e) 3π
2 f) 2π (d) 4 − √12 − √12 1
√ √
(e) 4π
− √23 − 12 3
55. a) 45◦ b) 60◦
3 √
(f) 5π
3 − 23 1
2 − 3
c) 150◦ d) 180◦ (g) 7π
4 − √12 √1 −1
√2
11π 3
(h) − 12 − √13
56. a) 0.1471 b) −0.4822 6 2
√
3
c) 1.1504 d) 1.8374 62. a) −1 b) 2
√
3
e) 0.5676 c) − 2 d) − √23
1
57. a) The body’s speed after four seconds is 33.33 m · s−1 . e) 0 f) 2
g) 1 h) √1
b) The body’s initial speed is 22 m · s−1 . 3
√
c) The body’s greatest speed is 38 m · s−1 . i) 2
3
j) 0

22
63. π 5π a)
a) x = 3, 3

b) x = π 5π 7π sin 3A cos 3A
3, 3 , 3 LHS = −
sin A cos A
c) x = − 5π π π 5π 7π sin 3A cos A − cos 3A sin A
3 ,−3, 3, 3 , 3 =
sin A cos A
sin(3A − A)
1
=
64. a) Solve sin A = 2 for A ∈ [− 11π 10π
3 , 3 ].
sin A cos A
sin 2A
=
b) A ∈ {− 19π 11π 7π π 5π 13π 17π
6 ,− 6 ,− 6 , 6, 6 , 6 , 6 }
sin A cos A
2 sin A cos A
=
c) x ∈ {− 3π 5π π π π π 3π
7 , − 21 , − 7 , 21 , 7 , 3 , 7 }
sin A cos A
= 2

65. 7π 11π 19π 23π

a) θ = , , ,
12 12 12 12
π π 5π 7π 3π 11π
b) θ = , , , , ,
6 2 6 6 2 6
2π 5π
c) θ = ,
3 3

66. 7π 11π b)
a) x = ,
6 6
LHS = cos 4x
π 2π 4π 5π
b) x = 0, , , π, , , 2π = cos(2 × 2x)
3 3 3 3
= 2 cos2 (2x) − 1
π 2π 4π 5π
c) x = 0, , , π, , , 2π = 2(cos 2x)2 − 1
3 3 3 3
= 2(2 cos2 x − 1)2 − 1
= 2(4 cos4 x − 4 cos2 x + 1) − 1
π π
67. θ = − , 0, = 8 cos4 x − 8 cos2 x + 2 − 1
2 2
= 8 cos4 x − 8 cos2 x + 1
√ = RHS
5
68. cos( π5 ) = 1
4 + 4 .

69.
23
 c) π 5π
b) x = ,
4 4
LHS = (cot t + cosec t)2
π 5π

cos t 1
2 c) 4 <x< 4
= +
sin t sin t
 2 Non-examinable Section
cos t + 1 h π i  7π 
= 71. a) x ∈ 0, ∪ , 2π
sin t 4 4
(cos t + 1)2 
π 3π

= b) x ∈ ,
sin2 t 4 4
(cos t + 1)2
= 72.
h π i  π 2π   3π 7π   5π 5π   7π 
1 − cos2 t a) x ∈ 0, ∪ , ∪ , ∪ , ∪ , 2π
(cos t + 1)2 6 4 3 4 6 4 3 4
= h π   3π 9π   11π 
(1 + cos t)(1 − cos t) b) x ∈ 0, ∪ , ∪ , 2π
(1 + cos t)2 8 8 8 8
= h π
(1 + cos t)(1 − cos t) 73. a) θ ∈ 0, ∪ (π, 2π]
1 + cos t 3 
= 3π

1 − cos t b) θ ∈ [0, π] ∪ , 2π
2
h πi        
π 5π 3π 9π 5π 13π 7π
c) θ ∈ 0, ∪ , ∪ , ∪ , ∪ , 2π
8 4 8 4 8 4 8 4
d) omitted
74. a)
e) omitted
f) omitted
•
70. a)
1 3
y
• A
y = cos x
1
√
2 2 √
b) cos A = − cot A = 2 2
π 3π x 3
2 π 2 2π √
75. a) 2 + 3
−1
y = sin x b) 3x − 4x3
c) Omitted. Hint: (a)6 = (a2 )3

24
76. Chapter 4

y Pre-tutorial Questions
84. a) x7 y 5
b) 288x8
√ √ √ √
(− 5π
3
, (− 2π
3) ◦ 3
, 3) ◦ ( π3 , 3) ◦ ( 4π
3
, 3) ◦ c) −2a15

• • • • • θ m4
d)
−2π −π O π 2π 16n4
85. a) 27n
b) 2n+1 3n+4

(θ = − 3π ) (θ = − π2 ) (θ = π
) (θ = 3π
) 1 1 1
2 2 2 86. (a) 32 (b) (c) 4 (d) (e)
3 125 16
77. 7π 3π 19π 7π 87. a) m 15
31
a) x ∈ [0, )∪( , ) ∪ ( , 2π]
12 4 12 4 7
3 15 − 35
7π 23π 31π 47π 55π 71π b) 2a b
b) t ∈ [ , ]∪[ , ]∪[ , ]
36 36 36 36 36 36
88. a) 0
3π
c) x ∈ [0, π] ∪ ( , 2π] b) 0
2
d) t ∈ (13, 21) ∪ (25, 33) ∪ (37, 40] c) −4.5
2 3
d)
e) θ ∈ [−6π, 2π] \ { π} ; alt.form θ ∈ [−6π, 32 π) ∪ ( 23 π, 2π] 2
3 e) 2 log10 a
f) 5a − 2
   
2π 14π
78. x ∈ −2π, ∪ 2π,
3 3
1
    89. (a) 2
5π π π 5π 3
79. θ ∈ − , − ∪ , (b) 2
6 6 6 6
Examinable Section
80. θ = −0.93, 0.49, 1.23
90. a) 4 log3 x + log3 (x + 3)
81. Answer omitted.
b) 3 log8 x − log8 (x − 8)
82. Omitted 1
c) ln a − 2 ln b − 5 ln c
83. Omitted 2
91.
25
 a) 2x 99. Graph omitted. Solution: x ∈ (0, 16).
b) 3y − 1 100. a) x ∈ (−3, 22]
c) 7 b) x ∈ (0, 41 ]
p
d) m
e) 5x Non-examinable Section
101. Omitted.
92. a) x = −3
102. Omitted.
b) x = 2
c) x = −2 Chapter 5
1
d) x = − Examinable Section
5
103. (a) No
93. a) x = 0
(b) No
b) x = 0 or x = 3
(c) No
94. a) x = 5 or x = −2 (d) Yes
b) x = 0.5 104. (a) dom(f ) = R \ {−4}
(b) dom(f ) = (−∞, 2]
95. a) x = 1.54 (2 d.p.)
(c) dom(f ) = R \ {±1}
b) x = −0.82 (2 d.p.)
(d) dom(f ) = [−1, 1]
96. a) x ∈ (−∞, 3] (e) dom(f ) = (−1, 1)
b) x ∈ [−2, ∞) (f) dom(f ) = R
c) x ∈ R 105. (a) dom(f ) = ( 21 , ∞)
d) x ∈ ∅ (b) dom(f ) = (−∞, 1)
e) x ∈ (−∞, 0) (c) dom(f ) = (1, ∞)

97. a) x ∈ (−∞, 14 ) 106. (a) dom(f ) = R \ {0}

(b) dom(f ) = R \ { 21 }
b) x ∈ ( 45 , ∞)
(c) dom(f ) = R
c) x ∈ [− 74 , ∞)
(d) dom(f ) = R
98. Graph omitted. Solution: x ∈ (8, ∞) (e) dom(f ) = [0, ∞)

26
 √
(f) dom(f ) = (−∞, 0] • (f − g)(x) = x2 − 2x + 1 − x
(g) dom(f ) = (−∞, −1] ∪ [1, ∞) dom(f − g) = [0, ∞)

(h) dom(f ) = − 12 , ∞ √


5 3
• (f g)(x) = x 2 − 2x 2 + x
(i) dom(f ) = (0, ∞) dom(f g) = [0, ∞)
(j) dom(f ) = (−∞, 1)   2

(k) dom(f ) = R \ {1, 3} • fg (x) = x −2x+1

√
x
 
f
x 1 dom g = (0, ∞)
107. (a) • (f + g)(x) = x−4 + 2x+3
√
dom(f + g) = R \ {4, − 32 } (d) • (f + g)(x) = 9 − x2 + 2x
dom(f + g) = [−3, 3]
x 1
• (f − g)(x) = x−4 − 2x+3 √
dom(f − g) = R \ {4, − 32 } • (f − g)(x) = 9 − x2 − 2x
dom(f − g) = [−3, 3]
x
• (f g)(x) = (x−4)(2x+3) √
dom(f g) = R \ {4, − 32 } • (f g)(x) = 2x 9 − x2
dom(f g) = [−3, 3]
 
• fg (x) = x(2x+3)   √
  x−4
2
• fg (x) = 9−x 2x
dom fg = R \ {4, − 32 }  
dom fg = [−3, 0) ∪ (0, 3]
√
(b) • (f + g)(x) = log2 (x + 2) + x21−1 (e) • (f + g)(x) = log3 x + 1 − x
dom(f + g) = (−2, ∞) \ {±1} dom(f + g) = (0, 1]
√
• (f − g)(x) = log2 (x + 2) − x21−1 • (f − g)(x) = log3 x − 1−x
dom(f − g) = (−2, ∞) \ {±1} dom(f − g) = (0, 1]
√
• (f g)(x) = logx22(x+2)
−1 • (f g)(x) = (log3 x) 1 − x
dom(f g) = (−2, ∞) \ {±1} dom(f g) = (0, 1]
   
• fg (x) = (x2 − 1) log2 (x + 2) • fg (x) = √ log3 x
    1−x
dom fg = (−2, ∞) \ {±1} dom fg = (0, 1)
√
(c) • (f + g)(x) = x2 − 2x + 1 + x 108. dom(f ) = [−5, 2] ∪ [3, ∞)
dom(f + g) = [0, ∞)
109. (a) f (g(x)) = (2x + 4)2

27
 (b) g(f (x)) = 2x2 + 4 115. dom(f ) = R \ { 95 }
9x − 3
110. (a) [−1, ∞) f −1 (x) =
5x − 2
(b) [1, ∞) ran(f −1 ) = R \ { 95 }
(c) [1, ∞) 116. (b), (c) and (d)
(d) [−1, ∞)
117. a) Omitted
(e) f −1 (x) = x2 − 2x
b) dom(f ) = [−2, 2] and ran(f ) = [0, 2]
(f)
y c) Since f is not one–one, f does not have an inverse function.
√
118. (a) b = 0, f −1 (x) = x+4
−1
√
(3, 3) (b) b = −2, f (x) = − x − 2
2
y = f (x) 119. (a) Graph omitted
f −1 (x) = − √1x − 2
(−1, 1) dom(f −1 ) = (0, ∞)
ran(f −1 ) = (−∞, −2)
2 x
(1, −1)
(b) Graph omitted
y = f −1 (x) g −1 (x) = √1x − 2
dom(g −1 ) = (0, ∞)
111. (a) dom(f ◦ g) = (−∞, −2) ran(g −1 ) = (−2, ∞)
(b) f (g(x)) = |x|
√
120. f −1 (x) = − 4 − x

112. (a) f ◦ g(x) = |x| 121. (a) b = −1

(b) dom(f ◦ g) = (−∞, −1] ∪ [1, ∞) (b) dom(f ) = (−∞, −1] ran(f ) = [−1, ∞)
−1
(c) dom(f ) = [−1, ∞) ran(f −1 ) = (−∞, −1]
4x2 − 2 if x > 12

113. f (g(x)) = −1
√
1 − 2x if x ≤ 12 (d) f (x) = −1 − x + 1

114. a) g ◦ f : [ 12 , 10] → R, g(f (x)) = 6x + 2 122. S = [−1, ∞)

b) f ◦ g : [4, 17
3 ] → R, f (g(x)) = 6x − 11
c) g ◦ g : [4, 35
3 ] → R, g(g(x)) = 9x − 28
d) f ◦ f : (−4, 72 ] → R, f (f (x)) = 4x + 9

28