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 in to 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 contain 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 examining 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 the anything in this booklet, contact your tutor. We hope you enjoy
your time in Maths 1.
The Mathematics 1 Team.
1
8. Decide whether the following statements are
true. For any that are false, provide an example where the statement in question does not
hold.
Chapter 1
a) A ∩ (B ∪ C) = (A ∩ B) ∪ C
Pre-Tutorial Questions
b) A ∩ (B ∩ C) = (A ∩ B) ∩ C
There are no pre-tutorial questions for this chapter.
c) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
9. The set A = {1, 2} has four subsets. List them.
Examinable Section
10. Use a number line to represent the following
intervals A, B, C, and D:
1. How would you read the following in English? Explain the meaning for each.
a) x ∈ A
b) A ⊆ B
c) A ∩ B
d) A ∪ B
e) A \ B
f) ∅
A = [0, 3), B = (−3, ∞), C = (−∞, 1], and
D = [−2, 2].
Hence find the following:
a) A ∩ B
b) A ∩ C
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
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. Let A = {1, 2, 3, 4, 5}. For each of the sets below, calculate A ∩ B, A ∪ B and A \ B.
3. State one element of each of the following
sets.
11. State the domain, codomain and range for
each of the following functions:
4. Is the set {∅} empty? Explain.
5. If A ⊆ B and B ⊆ A, what conclusion can you
draw?
a) f : R → R,
f (x) = 2x − 1
b) f : [0, ∞) → R,
6. For the sets A = {1, 2} and B = {2, 3}, select
which of the following statements are correct:
c) f : R+ → R,
f (x) = 2x − 1
f (x) = 2x − 1
f (x) = x2 − 1
a) 2 ∈ A
b) 2 ⊆ A
d) f : (−∞, 0] → R,
c) 2 = A ∩ B
d) {2} ∈ A
e) f : R− → R,
e) {2} ⊆ A
f) {2} = A ∩ B
f) f : [−3, 2) → R,
f (x) = 2x − 1
g) f : [−3, 2) → R,
f (x) = x2 − 1
h) f : R \ {0} → R,
f (x) = 2x − 1
7. Determine the value(s) of x ∈ Z, if any, for
which {2x, (2x)2 } contains two elements.
2
f (x) = 2x − 1
12. Consider the function
y
f : [−4, 3] → R where f (x) = x2 .
32
a) State the domain of f .
b) Find f (−4)
16
c) Find f (3)
d) Sketch the graph of y = f (x).
−4
−2
x
2
3
y = x − 12x + 16
e) State the range of f .
13. Consider the function
g : [−2, 2] → R where g(x) = −x2 .
State the domain and range of the following
functions:
(a) State the domain of g.
a) f : R → R given by
f (x) = x3 − 12x + 16.
(b) Find g(−2)
(c) Find g(2)
b) g : [−4, ∞) → R given by
g(x) = x3 − 12x + 16.
(d) Sketch the graph of y = g(x).
(e) State the range of g.
c) h : [−3, 3] → R given by
h(x) = x3 − 12x + 16.
14. Consider the function
h : [−2, 2] → R where h(x) =
p
4 − x2
d) p : [−3, 5] → R given by
p(x) = x3 − 12x + 16.
whose graph is shown here:
Non-examinable Section
y
Investigation Questions
16. The power set of a set S is defined as the set
of all subsets of S. It is denoted P(S).
2
a) Write the power set of B = {1, 2, 3}.
How many elements does it have?
−2
y=
√
2
4 − x2
b) If the set C has n elements, how many
elements does P(C) have? Justify your
answer.
x
a) State the domain of h.
b) State the range of h.
15. Consider the graph of y = x3 −12x+16 shown
here:
3
(a) 1 + x < 7x + 5
(b) 4 ≤ 3x − 2 < 13
Chapter 2
(c) x2 + 3x < 4
25. Solve the following inequalities for x:
Pre-tutorial Questions
1+x
>1
1−x
x
(b)
<4
3+x
(a)
Tutorial 2
17. Solve the following quadratic equations for x:
(a) x2 + 4x + 3 = 0
26. Solve for x:
2
(b) x + 13x + 42 = 0
(a) |3x − 6| = 6
(b) |x − 1| ≤ 3
18. Factorise the following expressions:
(c) |6x + 1| > 7
(a) x2 + 4x + 3
(b) x2 + 13x + 42
Examinable Section
(c) x3 + 4x2 + 3x
3
2
(d) x − 2x − x + 2
27. Solve the following equations for x
(a) 6x2 − 5x − 6 = 0
19. Solve for x:
(a) 2x3 − 4x2 + 2 = 0
(b) 2x2 + 3x + 1 = 0
(b) x4 − 10x2 + 9 = 0
(c) x2 + 9x − 10 = 0
(d) x2 + 9x − 1 = 0
20. Solve for x:
√
(a) x = x
√
(b) x = − x
(e) 2x2 − 2x − 2 = 0
(f) x2 + x + 1 = 0
(g) x2 + 2x + 1 = 0
(h) x2 − 2x − 2 = 0
21. Complete the square for the following
quadratic expressions:
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.
2
(a) x + 4x + 7
(b) 3x2 + 6x + 4
Tutorial 3
(a) 2x2 − 7x = −4
22. Factorise:
(b) 3x − x2 = 4
25
= 10
(c) 3x +
3x
(a) t3 + 1
(b) x2 − 4y 2
29. Factorise these expressions over R:
23. Find the expanded form of each of the following expressions:
(a) x2 + 2x + 1
(b) x2 − 2x − 2
(a) (a + x)6
(c) x2 − 3
(b) (a − x)6
(d) t2 − 6t + 2
(e) 12x2 + 5x − 3
24. Solve the following inequalities for x:
4
(f) 3x2 y 4 + 6x3 y 3
(c) 2x3 − 4x2 − 146x − 140
2
(d) p3 − 13p2 + 49p − 54
(g) x − 1
(e) x3 − 27
30. Complete the square for the following
quadratics:
(f) x3 + 2x2 − 5x − 6
(g) x3 + 2x2 − x − 2
(a) x2 − 4x + 7
(h) x4 − 2x2 − 3x − 2
(b) x2 − 6x + 10
(i) t3 + 3t2 + 3t + 126
(c) x2 + 4x − 3
(j) x4 − 1
(d) 2x2 + 8x − 6
(k) x3 − 6x2 + 12x − 8
(e) 3x2 + 3x + 1
(l) x3 y − y 3 x
31. Solve for x:
√
(a) 2x − 2 = x − 1
√
(b) 4x + 1 = 3 − 3x
√
(c) x + 1 = 1 − x
√
(d) 3x − 5 = x − 1
√
√
(e) 2x + 1 − x = 1
√
(f) 3x − 2 = −x
36. Use Pascal’s Triangle to find the expanded
form of each of the following expressions:
(a) (a + x)3
(b) (a + x)4
(c) (1 + x)3
(d) (5 − 2m)6
(e) ( 41 + 3x2 )5
1 4
) .
(f) (3t − 5t
32. Solve the following equations for their unknowns:
√
(a) 4k + 5 − 12 k = 2
√
(b) 1 + 2 − 3x = x.
(g) (1 + x2 )4
(h) (2 − x)4
37. Solve the following inequalities for x:
33. Use long division to divide
3
(a) 2x + 1 ≤ 4x − 3 ≤ x + 7
(b) x2 + 5x > −6,
2
(a) x + 2x + 2x + 2 by x + 1
(b) 2x3 + 9x2 + 10x + 23 by x + 4
38. Solve the following inequations for x. Write
your answers using the bracket notation for
intervals.
(c) x5 + 3x2 + 2x + 1 by x2 + 1
34. Solve for x:
(a) (x + 3)(3x − 2)(1 − x) ≥ 0
(a) x3 + 3x2 + 3x + 2 = 0
(b) (x − 1)(x − 2)(3 + 2x) ≤ 0
(b) x3 + 4x2 + 5x + 6 = 0
(c) (x2 + x + 1)(x − 1)(x + 2) < 0
(c) x4 − 2x2 − 3x − 2 = 0
(d) (x2 − 4)(2x + 1) < 0
(d) 2x3 − 9x2 + 9x − 2 = 0
(e) (x3 − 1)(x2 − 9) ≥ 0
(e) 2x3 − 7x2 − 3x + 18 = 0
39. Solve the following inequations for x. Write
your answers using the bracket notation for
intervals.
(f) x4 + 10x3 + 35x2 + 50x + 24 = 0
(g) x4 + 4x3 + 6x2 + 4x + 1 = 0
35. Factorise these expressions over R:
(a) 2x3 − 9x2 + 13x − 6 ≥ 0
(a) 2m3 − 6m2 + 2m + 2.
(b) −2x3 − x2 + 5x − 2 < 0
(b) 2w3 + 30w2 + 50w − 474
(c) 12x3 + 4x2 − 9x − 3 > 0
5
Non-examinable Section
(d) −2x3 + 12 ≤ −8x + 3x2
(e) x4 + 3x3 + 3x2 − x − 6 > 0
43. Show that all of the roots of x3 +2x2 −5x−3 =
0 are irrational.
40. Solve the following inequations for x. Write
your answers using the bracket notation for
intervals.
44. Fully factorise each expression over R:
(a) x4 − 7x2 + 6
2
≥2
2x + 3
x+1
(b)
<3
3x − 5
x2 + x − 6
(c)
≤0
4 + 3x
3x − 1 x + 1
1 − 2x
(d)
−
≥
2
4
3
2x + 3 3x + 1
−
≤ −3
(e)
2
3x
(a)
(b) x4 − x2 + 9.
(c) x4 + 64
(d) 9x4 + 2x2 + 1
(e) x8 − 1.
(f) x4 + x2 + 4
(g) 36x4 + 15x2 + 4
(h) t4 − t2 + 1
(i) t6 + 1
41. Solve for x:
45. Consider the equation
(a) |3x + 7| = 4
20x3 + 193x2 + 290x = 126.
(b) | − 2x − 3| = 2
(c) |2x + 3| = x + 2
Use a calculator or other technology to find
the roots, and write down each of them to 2
decimal places. Then use algebra to solve the
equation for x, giving the exact value of each
root.
(d) | − 2x + 5| = 3 − x
(e)
x+1
=3
2x + 1
42. Solve the following inequations for x:
46. Determine the values of x ∈ R that satisfy the
following inequations. Plot these solutions on
separate number lines.
(a) |7x + 2| ≥ 4
4
(b) |3x + 5| ≤
7
(c) | − 4x − 5| < |1 − 3x|
(a) −1 < x2 + 2x + 1 < 2
(b) x2 + 3x + 2 < 0
(d) |4x + 5| ≥ | − 1 + 3x|
(e)
(c) x(x − 4) > 5
4x + 3
≥1
−2 − 5x
(d) 0 < x2 − 7x + 10 < 1
(e) −1 < x2 + 9x + 19 < 0
5
1
(f)
>
−2 + x
1+x
(f) −3 < x2 + 7x + 9 < 9
47. Solve 3x2 −2x−1 ≤ 0 on Z (the set of integers).
48. Solve the following inequalities for x and
write your answers in interval notation:
(a) |x2 − 2| − x < 0
(b) |x2 − 6x + 6| < 2
6
Examinable Section
Chapter 3
54. Convert the following angles from degrees to
radians:
Pre-tutorial Questions
49. Convert the following angles from degrees to
radians:
b) 90◦
c) 120◦
d) 135◦
e) 270◦
f) 360◦
55. Convert the following angles from radians to
degrees:
a) 45◦
b) 60◦
π
radians
4
5π
c)
radians
6
a)
c) 150◦
50. Convert the following angles from radians to
degrees:
a)
a) 30◦
b)
π
radians
3
d) π radians
56. Using your calculator find the value (to 4 decimal places) of the following:
3π
2 rad
b) 2π
3 rad
51. Using your calculator write the following o
four decimal places:
a) cos 4.86
b) sin 5.78
c) tan 49◦
d) cot 3.64
e) cos 5.316
a) sin 53◦
57. A body moves so that its speed, V m·s−1 , after
t seconds is given by
t
.
V = 30 − 8 cos
2
b) cot 0.4
52. Without using a calculator, find the exact values of the expressions given below:
a) cosec π2
a) Calculate the body’s speed after four
seconds. Present your answer in a sentence and to two decimal places.
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.
b) Calculate the body’s initial speed. (That
is, its speed when t = 0.)
a) The angle θ between the x-axis and the
ray extending
from the origin to the
√
point (1, 3).
c) Calculate the body’s greatest speed.
(Do not use calculus to answer this
question.)
b) The angle θ between the x-axis and the
ray extending
from the origin to the
√
point ( 3, 1).
58. The depth of water in a particular part of a
bay varies with time according to the following formula:
π D = 20 + 3 sin
t
12
where D is the depth measured in metres, and
t is the number of hours after midnight, with
0 ≤ t ≤ 24.
7
a) Find the depth when t = 0.
7π
4
5π
3
a) cot
b) Find the depth at 1 am. Write your answer to 2 decimal places.
c) sin
c) Find the depth at 2 am.
e) Find the minimum depth.
i) cos
b) cos(2π − θ)
π
d) cos
−θ
2
63.
b)
5π
6
c)
7π
6
d)
5π
4
13π
6
2
+θ
4π
e)
3
5π
f)
3
7π
g)
4
11π
h)
6
a) Solve cos x = 21 for 0 ≤ x ≤ 2π.
c) Solve cos x = 21 for −2π ≤ x ≤ 3π.
64. Solve
h π πi
π 1
= for x ∈ − ,
sin 7x −
6
2
2 2
using the following steps:
a) Rewrite the equation in terms of a new
unknown A = 7x − π6 .
b) Solve the new equation for A, writing
the values in a set from least to greatest.
c) Hence complete the original task.
65. Solve the following equations for θ ∈ [0, 2π]:
1
2
π
a) sin (2θ) = −
61. For each of the following angles θ, find exact
values of sin θ, cos θ and tan θ without the use
of a calculator.
3π
4
60. Without using a calculator, find the exact values of the expressions given below:
2π
5π
a) sin
b) cot
3
6
5π
7π
c) tan
d) sin
4
6
7π
3π
e) cos
f) sec
6
4
5π
4π
g) sec
h) cot
4
3
a)
9π
4
b) Solve cos x = 21 for 0 ≤ x ≤ 3π.
f) tan(π − θ)
π
g) tan
59. Express each of the following in terms of
sin θ, cos θ or tan θ
h) sin
11π
b) cos
6
5π
d) cosec
3
13π
f) sin
6
7π
h) cot
3
5π
j) cos
2
e) sin (2π)
d) Find the maximum depth.
a) sin(2π − θ)
π
c) sin
−θ
2
π
e) sec
−θ
2
π
g) cos
+θ
2
b) tan 3θ +
c) sin θ +
√
4
= −1
3 cos θ = 0
66. Solve the following for x ∈ [0, 2π]:
a) sin x = −
1
2
b) sin(3x) = 0
√
π
c) 3 tan 3x −
= −1
6
h π πi
π
67. Solve tan 2θ +
= 1 for θ ∈ − , .
4
2 2
62. Without using a calculator, find the exact values of the expressions given below:
8
68. Given that
74. Suppose that A is the angle in −π, − π2 for
which cosec A = −3.
√
5−1
4
find the exact value of cos π5 .
sin
π
10
=
a) Draw a diagram that shows the angle A.
b) Evaluate cos A and cot A.
69. Prove the following identities:
75.
sin(3A) cos(3A)
−
=2
sin A
cos A
b) cos(4x) = 8 cos4 x − 8 cos2 x + 1
a)
c) (cot t + cosec t)2 =
1 + cos t
1 − cos t
5π
.
12
π
3π
. Express sin
b) Let x be sin
in
7
7
terms of x.
a) Determine the exact value of tan
c) Prove the following identity:
1
1
d)
+
= 2 sec2 θ
1 − sin θ 1 + sin θ
e) (1 − tan θ)2 + (1 + tan θ)2 = 2 sec2 θ
cos6 t + sin6 t = 1 − 3 cos2 t sin2 t.
f) sin(α + β) + sin(α − β) = 2 sin α cos β
70.
76. With θ as the variable on the horizontal axis,
sketch the graph of y = tan(θ) from θ = −2π
to θ = 2π. On your graph,
highlight the points
√
for which tan(θ) < 3.
a) Using the same set of axes, sketch the
graphs of y = sin x and y = cos x between x = 0 and x = 2π.
b) Find the x–coordinates of the points of
intersection of the two graphs.
77. Solve each of the following inequations:
√
π
a) 2 cos 2x −
> − 3 for x ∈ [0, 2π]
3
π
b) 2 sin 3t +
≤ 1 for t, where 0 ≤ t ≤
4
2π
x π
c) tan
−
≤ 1 for x, where 0 ≤ x ≤
2
4
2π
πt π
−
d) 2 sin
+4 > 3 for t, where 12 <
6
3
t ≤ 40
θ π
e) cos
−
< 1 for θ ∈ [−6π, 2π].
4
6
x π
8π 16π
78. Solve sin
−
> 12 for x ∈ − ,
.
4
3
3
3
c) Hence find the values of x between 0
and 2π for which sin x > cos x.
Non-Examinable Section
71. Solve each of the following inequalities, for
x ∈ [0, 2π]:
1
a) cos x − √ ≥ 0
2
1
b) sin x − √ ≥ 0
2
72.
√
a) Solve tan(2x) − 3 ≤ 0 for x ∈ [0, 2π].
√
b) Solve 2 sin(2x − π) + 1 > 0 for x ∈
[0, 2π].
73. Solve the following inequalities, for θ
[0, 2π]:
π 1
a) sin θ −
− <0
6
2
√
π
b) 2 cos θ −
+1≥0
4
π
c) tan 2 θ +
≤1
2
∈
79. Solve cos4 θ − sin4 θ <
1
for θ ∈ [−π, π].
2
80. Find, to two decimal
all real values of
π places,
π
θ in the interval − ,
for which tan(θ) +
2 2
tan(2θ) = 2 by substituting x for tan(θ) and
then using a calculator or other technology to
solve the resulting cubic equation.
9
81. Determine the exact value of 1 − sin
π
16
x1
.
x2
82. One way to prove the two double angle formulas for 2θ ∈ (0, π2 ) is by 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 edge of
the circle. Then:
• the angle ∠COB is twice that of the angle ∠CAB.
y2
y
π
2
1
• the angle ∠ACB is a right angle.
Consider the following picture:
y1
C
α
β
α
x
A
θ
2θ
O
B
Calculate each of the unknown lengths in
terms of sin α, cos α, sin β and cos β. Then
compare the lengths of the opposite sides of
the rectangle.
Using these facts above, and the definitions of
sin and cos.
a) Find the value of α
b) Identify the length corresponding to
sin(2θ) and calculate it in terms of cos θ
and sin(θ).
c) Identify the length corresponding to
cos(2θ) and calculate it in terms of cos θ
and sin(θ).
(Hint: Make your life easier by making the circle in your picture a unit circle.)
83. One geometric proof of the addition formulae
involves setting up the following rectangle:
10
89. Simplify the following.
Chapter 4
Pre-tutorial exercises
(a)
log2 9
log2 81
(b)
2 log3 8
log3 16
84. Simplify the following expressions.
a) x5 y × x2 y 4
2
Examinable Section
2 3
b) (6x) × (2x )
90. Write the following as sums and/or differences of logarithms.
c) (−a3 )5 × 2(a−3 )0
m 2 2n −2
d)
2n
m
a) log3 (x4 (x + 3)) where x ∈ (0, ∞)
85. Simplify the following expressions.
4n
a) 2
b)
3n
×4
b) log8
−n
×8
63n × 9n+2 × 8
81n × 4n+1
c) ln
86. Evaluate the following without using your
calculator.
x3
x−8
where x ∈ (8, ∞)
√ a
where a, b, c ∈ (0, ∞)
b2 c5
91. Use exponential and logarithmic laws, and
the fact that aloga (x) = x (where a ∈ R+ \ {1},
x > 0), to simplify the following expressions.
a) 25
1
b) 9− 2
2
c) 32 5
d) 25
a) eln(2x) where x ∈ (0, ∞)
− 23
b) ln(e3y−1 )
− 23
e) (−64)
c) ln(e2 e5 )
87. Simplify the following expressions.
√
√
5
3
a) m2 × m5
√
√
3
5
b) 27a2 b0 ÷ 32ab3
d) ep ln(m) where m ∈ (0, ∞)
e) eln x+ln 5 where x ∈ (0, ∞)
88. Simplify the following to a single logarithm
and evaluate where possible without using a
calculator.
92. Solve the following exponential equations for
x ∈ R.
a) log10 2 + log10 3 − log10 6
a) 2x = 0.125
b) 4 log10 3 − log10 81
1
1
c) log3
− log7 49 + log5 √
9
5
log10 8
d)
log10 4
√
5
log10 a − log10 a where a ∈ (0, ∞)
e)
2
f) 3log3 (5a−2) where a ∈ ( 25 , ∞)
b) 4x = 83−x × 2x−1
c) 92x+1 = 27x
d)
11
3x−1
=1
272x
Non-Examinable Section
93. Solve the following equations for x ∈ R.
(Hint: Use substitution)
Challenge
a) 22x = 2x+1 − 1
x
x
b) 4 − 9(2 ) + 8 = 0
101.
94. Solve each of the following for x ∈ R.
a) ln(x2 − 3x − 9) = 0
(a) Show that for all values of the parameter
c, the equation
1 x
=c
2
!
x
2 −
b) log3 (5x + 11) − log3 x = 3
95. Solve for x ∈ R, giving your answers correct
to 2 decimal places.
x
a) 5 = 12
b) 3−2x = 6
has one solution for the unknown x.
(b) Suppose that 2x +2−x = c. For each value
of c, determine the number of solutions
for x.
96. Sketch the graph of y = 2 x . Remember to 102. Solve the following inequalities for x ∈ R.
show the asymptote and to include a label for
a) ln(x + 3) + ln(x + 5) > 2
the intercept(s).
b) log4 (1 + x) − log4 (1 − x) ≤ 0
Then solve the following inequations for x ∈
R.
a) 2x ≤ 8
b) 2x ≥ 0.25
c) 2x > −3
d) 2x < −3
e) 2x < 1
97. Solve for x ∈ R in each of the following.
a) 16x < 2
b) 0.54x−1 <
c) 42x+1 ≥
1
16
1
32
98. Sketch the graph of y = log2 x. Remember to
label the x–intercept and the asymptote. Label two more points on the curve: (8, 3) and
(x, y) for a general x greater than 8. Hence
write down the solution to log2 x > 3.
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 solution to log4 x < 2.
100. Solve for x ∈ R.
a) log5 (x + 3) ≤ 2
b) 5 + 3 log0.5 x ≥ 11
12
(c) f (x) = 2x − 1
(d) f (x) = sin x
√
(e) f (x) = x
√
(f) f (x) = −x
√
(g) f (x) = x2 − 1
Chapter 5
Pre-tutorial exercises
(h) f (x) = log4 (2x + 1)
1
(i) f (x) = √
x
x
(j) f (x) = √
1−x
1
(k) f (x) = 2
x − 4x + 3
Tutorial 6
There are no pre-tutorial questions for this week
Tutorial 7
103. Are the following functions one–one?
(a) f : [0, 2π] −→ R where f (x) = sin x
(b) f : [0, π] −→ R where f (x) = sin x
(c) f : [0, 2π] −→ R where f (x) = cos x
107. In each of the following, find the rules for
f
f + g, f − g, f g and , and state the correg
sponding domains:
(d) f : [0, π] −→ R where f (x) = cos x
(a) f (x) =
Examinable Section
(b) f (x) = log2 (x + 2) and g(x) =
104. Find the implied domain for the following
functions:
4x + 1
(a) f (x) =
x+4
√
(b) f (x) = 2 − x
4
(c) f (x) =
1 − x2
√
(d) f (x) = 1 − x2
4x
(e) f (x) = √
1 − x2
(f) f (x) = 1 + x2
(c) f (x) = x2 − 2x + 1 and g(x) =
(d) f (x) =
x
2x − 1
x
9 − x2 and g(x) = 2x
√
1−x
(a) find the rule for f (g(x)).
(b) find the rule for g(f (x)).
110. Consider the function f (x) = 1 +
√
x + 1.
(a) Find dom(f ).
(b) Find ran(f ).
(c) Find dom(f −1 ).
106. Find the implied domain of the following
functions:
(b) f (x) =
√
109. If f (x) = x2 and g(x) = 2x + 4 then
(c) f (x) = 2 log4 (x − 1)
1
x
1
x2 − 1
108. Find the implied domain of the function
p
f (x) = x3 − 19x + 30.
(b) f (x) = log10 (1 − x)
(a) f (x) =
√
(e) f (x) = log3 x and g(x) =
105. Find the implied domain of the following
functions:
(a) f (x) = log2 (2x − 1)
1
x
and g(x) =
x−4
2x + 3
(d) Find ran(f −1 ).
(e) Find the rule for f −1 .
(f) Sketch the graphs of y = f (x) and y =
f −1 (x).
13
111. Consider the functions
f : (4, ∞) −→ R where f (x) =
√
117. Consider the function given by
p
f (x) = 4 − x2 .
x
and
a) Sketch the graph of y = f (x).
−
2
g : R −→ R where g(x) = x .
b) Find the domain and range of f .
(a) Find the domain of f ◦ g.
c) Does f have an inverse function? Justify
your answer.
(b) Find the rule for f (g(x)).
112. Consider the functions
118.
f : [1, ∞) −→ R where f (x) =
√
x
and
g : R −→ R where g(x) = x2 .
(b) Find the largest number b such that the
function f : (−∞, b] → R given by
f (x) = (x + 2)2 has an inverse function.
Find the rule for the inverse function.
(a) Find the rule for f ◦ g.
(b) Find dom(f ◦ g).
2
x − 2 if x > 1
113. If f (x) =
and g(x) = 2x
1−x
if x ≤ 1
119.
then find the rule for f (g(x)).
f : (−5, 10] → R, f (x) = 2x + 3
g : [4, 28] → R, g(x) = 3x − 7
c) g ◦ g
d) f ◦ f
2x − 3
5x − 9
→
R given by
:
→
R given by
i. Sketch the graphs of y = g(x) and
y = g −1 (x).
ii. Completely determine g −1 . That is,
find the domain, range and rule for
g −1 .
115. This exercise is about the function
f (x) =
:
(−2, ∞)
1
.
g(x) =
(x + 2)2
(b) Let g
Use that same format to express the following
composite functions.
b) f ◦ g
(−∞, −2)
1
.
f (x) =
(x + 2)2
(a) Let f
i. Sketch the graphs of y = f (x) and
y = f −1 (x).
ii. Completely determine f −1 . That is,
find the domain, range and rule for
f −1 .
114. The functions f and g below have been expressed in a particular format.
a) g ◦ f
(a) Find the smallest number b such that
the function f : [b, ∞) → R given by
f (x) = x2 − 4 has an inverse function.
Find the rule for the inverse function.
120. Find the rule for f −1 if f is the function defined by
that has maximal domain. Find the domain of
f, the rule for f −1 (x), and the range of f −1 .
f : [−2, 0] → R where f (x) = 4 − x2 .
116. Which of the following functions has an in121. Consider the function
verse function?
a) f : R → R where f (x) = x2 − 2
f : (−∞, b] → R where f (x) = x2 + 2x.
b) f : [0, ∞) → R where f (x) = x2 + 1
a) Find the largest value of b so that f has
an inverse function.
c) f : R → R where f (x) = x
3
d) f : [−1, ∞) → R where f (x) = (x + 3)2
Using this value of b,
14
Answers
b) state the domain and range of f .
c) state the domain and range of f −1 .
d) find f
−1
Chapter 1
.
Examinable Section
122. Consider the function
1. Find these terms in the course notes.
f : S → R where f (x) = 2x + 2.
2.
If f has inverse function given by
f −1 : [0, ∞) → R where f −1 (x) =
1
x − 1,
2
a) A ∩ B = {2, 4}
A ∪ B = {0, 1, 2, 3, 4, 5, 6, 8}
A \ B = {1, 3, 5}
b) A ∩ B = {1, 3, 5}
A ∪ B = {1, 2, 3, 4, 5}
A \ B = {2, 4}
then find the set S.
c) A ∩ B = {}
A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A\B =A
3. Omitted
4. {∅} is not empty, it contains a single element:
∅.
5. If A ⊆ B and B ⊆ A, then A = B.
6. a), e) and f) are correct.
7. x ∈ Z \ {0}.
8.
a) False. e.g. if A = {1, 2, 4, 5}, B =
{2, 3, 5, 6}, C = {4, 5, 6, 7}
b) True (This is called the associativity
property)
c) True (Compare this to the distributive
law for multiplication and addition)
9. ∅, {1}, {2}, {1, 2}
10. A number line representing the sets, A, B, and
C is shown below.
D•
•
• C
B
•
−4 −3 −2 −1 0
15
A
x
1
2
3
4
a) [0, 3)
b) [0, 1]
a) dom(g) = [−2, 2]
c) [0, 2]
d) (−3, 1]
b) −4
e) [−2, 2]
f) [−2, 1]
g) [0, 1]
h) (−3, ∞)
i) (−∞, 3)
j) [−2, 3)
k) (−∞, ∞) = R
l) (−3, ∞)
m) (−∞, 2]
n) (−∞, ∞) = R
o) ∅
p) (−3, 0) ∪ [3, ∞)
q) (1, 3)
r) (−∞, 0)
s) (2, 3)
t) [−2, 0)
c) −4
d)
y
−2
2
x
−4
y = g(x)
e) ran(g) = [−4, 0].
14.
a) dom(h) = [−2, 2]
b) ran(h) = [0, 2]
11. The codomain for all these functions is R.
15.
a) dom f = R, ran f = R
a) dom(f ) = R and ran(f ) = R
b) dom(g) = [−4, ∞) and ran(g) = [0, ∞)
b) dom f = [0, ∞), ran f = [−1, ∞)
c) dom(h) = [−3, 3] and ran(h) = [0, 32]
c) dom f = R+ , ran f = (−1, ∞)
d) dom(p) = [−3, 5] and ran(p) = [0, 81]
d) dom f = (−∞, 0], ran f = [−1, ∞)
Non-examinable Section
e) dom f = R− , ran f = (−∞, −1)
16. P(B) has 8 elements:
∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, B
f) dom f = [−3, 2), ran f = [−7, 3)
If B has n elements, P(B) has 2n elements.
g) dom f = [−3, 2), ran f = [−1, 8]
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 sets).
h) dom f = R \ {0}, ran f = R \ {−1}
12.
a) dom(f ) = [−4, 3]
Chapter 2
b) 16
c) 9
Pre-tutorial Questions
d)
17.
y
(a) x = −1, −3
(b) x = −6, −7
16
18.
y = f (x)
9
(a) (x + 1)(x + 3)
(b) (x + 6)(x + 7)
(c) x(x + 1)(x + 3)
−4
(d) (x − 2)(x + 1)(x − 1)
3 x
√
19.
e) ran(f ) = [0, 16]
(a) x = 1, 1±2 5
(b) x = ±1, ±3
13.
16
20.
21.
22.
23.
(a) x = 0, 1
(f) 3x2 y 3 (y + 2x)
(b) x = 0
(g) (x + 1)(x − 1)
(a) (x + 2)2 + 3
30.
(b) 3[(x + 1)2 + 31 ] = 3(x + 1)2 + 1
(b) (x − 3)2 + 1
(a) (t + 1)(t2 − t + 1)
(c) (x + 2)2 − 7
(b) (x − 2y)(x + 2y)
(d) 2(x + 2)2 − 14
(a) a6 + 6a5 x + 15a4 x2 + 20a3 x3 + 15a2 x4 +
6ax5 + x6
6
5
4 2
3 3
(e) 3(x + 21 )2 + 14
31.
2 4
(c) x = 0
2
24. (a) x ∈ (− , ∞)
3
(b) x ∈ [2, 5)
(d) x = 2, 3
(e) x = 0, 4
(c) x ∈ (−4, 1)
32.
(a) x ∈ (0, 1)
33.
(a) x ∈ {4, 0}
(c) x3 − x + 3 + 3x−2
x2 +1
4
(c) x ∈ (−∞, − ) ∪ (1, ∞)
3
34.
Examinable Section
(b) x = −3
(c) x = −1, 2
(b) x = − 21 , −1
(d) x = 1, 7±4 33
(d)
(e)
√
(e) x = 2, − 32 , 3
√
x = −9±2 85
√
x = 1±2 5
(f) x = −1, −2, −3, −4
(g) x = −1
(f) No real solutions.
(g) x = −1
(h) x = 1 ±
29.
(a) x = −2
(a) x = 32 , − 23
(c) x = 1, −10
28.
1
(a) x2 + x + 1 + x+1
−1
(b) 2x2 + x + 6 + x+4
(b) x ∈ [−2, 4]
27.
(f) No solutions.
√
√
(a) k = −2 5 + 4, k = 2 5 + 4
(b) There are no possible (real) values for x.
(b) x ∈ (−∞, −4) ∪ (−3, ∞)
26.
(a) x = 1, 3
(b) x = 49
(b) a − 6a x + 15a x − 20a x + 15a x −
6ax5 + x6
25.
(a) (x − 2)2 + 3
√
35.
3
√
√
(a) 2(m − 1)(m − 1 − 2)(m − 1 + 2)
√
√
(b) 2(w − 3)(w + 9 − 2)(w + 9 + 2)
(c) 2(x + 7)(x + 1)(x − 10)
√
(a) 17, two distinct real roots
(d) (p − 2)(p − 11
2 −
(b) −7, no real roots
(e) (x − 3)(x2 + 3x + 9)
(c) 0, one real root
(f) (x − 2)(x + 1)(x + 3)
(a) (x + 1)
2
√
13
11
2 )(p − 2 +
(g) (x − 1)(x + 1)(x + 2)
√
(b) (x − 1 − 3)(x − 1 + 3)
√
√
(c) (x − 3)(x + 3)
√
√
(d) (t − 3 − 7)(t − 3 + 7)
(h) (x + 1)(x − 2)(x2 + x + 1)
(e) (3x − 1)(4x + 3)
(k) (x − 2)3
(i) (t + 6)(t2 − 3t + 21)
(j) (x2 + 1)(x + 1)(x − 1)
17
√
13
2 )
5 1
(b) x ∈ {− , − }
2 2
5
(c) x ∈ {− , −1}
3
8
(d) x ∈ {2, }
3
4 2
(e) x ∈ {− , − }
7 5
(l) xy(x + y)(x − y)
36.
(a) a3 + 3a2 x + 3ax2 + x3
(b) a4 + 4a3 x + 6a2 x2 + 4ax3 + x4
(c) 1 + 3x + 3x2 + x3
(d) 64m6 − 960m5 + 6000m4 − 20000m3 +
37500m2 − 37500m + 15625
135 6
45 4
15 2
8
(e) 243x10 + 405
4 x + 8 x + 32 x + 256 x +
1
1024
4
42.
54
12
1
2
(f) 81t − 108
5 t + 25 − 125t2 + 625t4
(g) 1 + 4x2 + 6x4 + 4x6 + x8
(h) 16 − 32x + 24x2 − 8x3 + x4
37.
38.
39.
40.
41.
1
(a) x ∈ [2, 3 ]
3
(b) x ∈ (−∞, −3) ∪ (−2, ∞)
2
(a) x ∈ (−∞, −3] ∪ [ , 1]
3
3
(b) x ∈ (−∞, − ] ∪ [1, 2]
2
(c) x ∈ (−2, 1)
1
(d) x ∈ (−∞, −2) ∪ (− , 2)
2
(e) x ∈ [−3, 1] ∪ [3, ∞)
Non-examinable Section
43. omitted
44.
3
(a) x ∈ [1, ] ∪ [2, ∞]
2
1
(b) x ∈ (−2, ) ∪ (1, ∞)
2
√
√
3 1
3
(c) x ∈ (−
,− ) ∪ (
, ∞)
2
3
2
3
(d) x ∈ [−2, − ] ∪ [2, ∞)
2
(e) x ∈ (−∞, −2) ∪ (1, ∞)
√
√
(a) (x + 1)(x − 1)(x + 6)(x − 6)
√
√
(b) (x2 + 7x + 3)(x2 − 7x + 3)
(c) (x2 − 4x + 8)(x2 + 4x + 8)
(d) (3x2 + 2x + 1)(3x2 − 2x + 1)
√
(e) √
(x + 1)(x − 1)(x2 + 1)(x2 + 2x + 1)(x2 −
2x + 1)
√
√
(f) (x2 − 3x + 2)(x2 + 3x + 2)
(g) (6x2 − 3x + 2)(6x2 + 3x + 2)
√
√
(h) (t2 − 3t + 1)(t2 + 3t + 1)
√
√
(i) (t2 + 1)(t2 − 3t + 1)(t2 + 3t + 1)
3
(a) x ∈ (− , −1]
2
5
(b) x ∈ (−∞, ) ∪ (2, ∞)
3
4
(c) x ∈ (−∞, −3] ∪ (− , 2)
3
13
(d) x ∈ [ , ∞)
23
√
√
−21 − 489
489 − 21
(e) x ∈ [−∞,
] ∪ (0,
]
12
12
(a) x ∈ {−
6
2
(a) x ∈ (−∞, − ] ∪ [ , ∞)
7
7
13 31
(b) x ∈ [− , − ]
7
21
4
(c) x ∈ (−6, − )
7
4
(d) x ∈ (−∞, −6] ∪ [− , ∞)
7
2
5 2
(e) x ∈ [− , − ) ∪ (− , 1]
9 5
5
1
7
(f) x ∈ (−∞, − ) ∪ (− , 2) ∪ (2, ∞)
4
2
(a) −7.65, −2.35, 0.35
√
√
7
(b) x1 = −5 − 7, x2 = −5 + 7, x3 = 20
or
x3 = 0.35
√
√
46. (a) −1 − 2 < x < −1 + 2
45.
(b) −2 < x < −1
(c) x > 5 or x < −1
√
√
(d) 5 < x < 7+2 13 or 7−2 13 < x < 2
√
√
(e) −9−2 5 < x < −5 or −4 < x < −9+2 5
11
, −1}
3
(f) −7 < x < −4 or −3 < x < 0
18
47. If we solve the inequality over the reals we
get − 31 ≤ x ≤ 1 the only elements of Z in that
interval are 0 and 1.
48.
a) The body’s speed after four seconds is
33.33 m · s−1 .
b) The body’s initial speed is 22 m · s−1 .
(a) x ∈ (1, 2)
√
√
(b) x ∈ (3 − 5, 2) ∪ (4, 3 + 5)
c) The body’s greatest speed is 38 m · s−1 .
Chapter 3
58.
b) At 1 am the depth is ≈ 20.78 m.
Pre-tutorial Questions
49.
a)
c) At 2 am the depth is 21.5 m.
π
4
d) The maximum depth is 23 m.
b) π3
c)
50.
e) The minimum depth is 17 m.
5π
6
59.
a) 270◦
b) 120
51.
◦
a) 0.7986
60.
b) 2.3652
52.
sin
θ
√
3
2
1
2
cos θ
1
√2
3
2
tan θ
√
3
cosecθ
√1
3
2
√2
3
sec θ
2
55.
56.
d) sin θ
e) cosecθ
f) − tan θ
g) − sin θ
h) cos θ
√
3
2
√
b) − 3
a)
√2
3
cot θ 61.
√1
√3
(a)
3
(b)
(c)
(d)
Examinable Section
54.
c) cos θ
d) − 21
√
f) − 2
√
c) 0
(b)
b) cos θ
e) − 23
√
g) − 2
b) 0
(a)
a) − sin θ
c) 1
a) 1
53.
a) When t = 0 the depth is 20 m.
h) √13
θ
sin θ
3π
4
5π
6
7π
6
5π
4
4π
3
5π
3
7π
4
11π
6
√1
2
1
2
− 12
− √12
√
− √23
− 23
− √12
− 12
cos θ
− √12
tan θ
−1
− 23
− √13
− 23
− √12
√1
3
√
√
1
√
3
√
− 3
−1
a)
π
6
b) π2
c)
2π
3
d) 3π
4
e)
3π
2
f) 2π
a) 45◦
b) 60◦
c) 150◦
d) 180◦
c) − 23
d) − √23
a) 0.1471
b) −0.4822
e) 0
f)
c) 1.1504
d) 1.8374
g) 1
h) √13
(e)
(f)
(g)
(h)
62.
− 21
1
2
1
√
√2
3
2
− √13
√
a) −1
√
b)
3
2
1
2
√
e) 0.5676
i)
63.
57.
19
3
2
j) 0
a) x = π3 , 5π
3
b)
LHS
7π
b) x = π3 , 5π
3 , 3
π π 5π 7π
c) x = − 5π
3 ,−3, 3, 3 , 3
64.
a)
10π
Solve sin A = 12 for A ∈ [− 11π
3 , 3 ].
11π
7π π 5π 13π 17π
b) A ∈ {− 19π
6 ,− 6 ,− 6 , 6, 6 , 6 , 6 }
5π
π π π π 3π
c) x ∈ {− 3π
7 , − 21 , − 7 , 21 , 7 , 3 , 7 }
=
cos 4x
=
cos(2 × 2x)
=
2 cos2 (2x) − 1
=
2(cos 2x)2 − 1
=
2(2 cos2 x − 1)2 − 1
=
2(4 cos4 x − 4 cos2 x + 1) − 1
=
8 cos4 x − 8 cos2 x + 2 − 1
=
8 cos4 x − 8 cos2 x + 1
= RHS
65.
66.
a) θ =
7π 11π 19π 23π
,
,
,
12 12 12 12
b) θ =
π π 5π 7π 3π 11π
, ,
,
,
,
6 2 6 6 2
6
c) θ =
2π 5π
,
3 3
c)
LHS
=
=
7π 11π
a) x =
,
6
6
=
b) x = 0,
π 2π
4π 5π
,
, π,
,
, 2π
3 3
3 3
=
c) x = 0,
π 2π
4π 5π
,
, π,
,
, 2π
3 3
3 3
=
π
π
67. θ = − , 0,
2
2
=
√
68. cos( π5 ) = 14 + 45 .
=
69.
=
a)
LHS
sin 3A cos 3A
−
sin A
cos A
sin 3A cos A − cos 3A sin A
=
sin A cos A
sin(3A − A)
=
sin A cos A
sin 2A
=
sin A cos A
2 sin A cos A
=
sin A cos A
= 2
=
d) omitted
e) omitted
f) omitted
20
70.
(cot t + cosec t)2
2
cos t
1
+
sin t
sin t
2
cos t + 1
sin t
(cos t + 1)2
sin2 t
(cos t + 1)2
1 − cos2 t
(cos t + 1)2
(1 + cos t)(1 − cos t)
(1 + cos t)2
(1 + cos t)(1 − cos t)
1 + cos t
1 − cos t
a)
75.
y
a) 2 +
√
3
b) 3x − 4x3
y = cos x
1
c) Omitted. Hint: (a)6 = (a2 )3
76.
π
2
3π
2
π
−1
2π
x
y
y = sin x
(− 5π
,
3
π 5π
b) x = ,
4 4
c)
74.
a)
77.
3) ◦(− 2π
,
3
√
3) ◦ ( π3 ,
•
−2π
−π
(θ = − 3π
)
2
(θ = − π2 )
5π
π
4 <x< 4
Non-Examinable Section
h π i 7π
71.
a) x ∈ 0,
, 2π
∪
4
4
π 3π
b) x ∈
,
4 4
h π i π 2π 3π 7π 72.
a) x ∈ 0,
,
∪
,
∪
∪
4 6
6 4 3
5π 5π
7π
,
, 2π
∪
4 3
4
h π 3π 9π 11π
b) x ∈ 0,
∪
,
, 2π
∪
8
8 8
8
h π
73.
a) θ ∈ 0,
∪ (π, 2π]
3 3π
b) θ ∈ [0, π] ∪
, 2π
2
h π i π 5π 3π 9π c) θ ∈ 0,
∪
,
∪
,
∪
8 4 8
4 8
5π 13π
7π
,
∪
, 2π
4
8
4
√
•
A
Pre-tutorial exercises
84.
21
(θ = 3π
)
2
2π
14π
−2π,
∪ 2π,
3
3
5π π
π 5π
79. θ ∈ − , −
∪
,
6
6
6 6
78. x ∈
83. Omitted
√
cot A = 2 2
(θ = π2 )
2
e) θ ∈ [−6π, 2π] \ { π} ; alt. form θ ∈
3
[−6π, 23 π) ∪ ( 23 π, 2π]
Chapter 4
√
2 2
b) cos A = −
3
π
d) t ∈ (13, 21) ∪ (25, 33) ∪ (37, 40]
82. Omitted
•
•
O
7π
3π 19π
7π
)∪( ,
) ∪ ( , 2π]
12
4 12
4
7π 23π
31π 47π
55π 71π
b) t ∈ [ ,
]∪[
,
]∪[
,
]
36 36
36 36
36 36
3π
c) x ∈ [0, π] ∪ ( , 2π]
2
81. Answer omitted.
3
1
•
a) x ∈ [0,
80. θ = −0.93, 0.49, 1.23
•
√
√
3) ◦ ( 4π
, 3) ◦
3
• θ
2π
a) x7 y 5
a) x = 0
b) 288x8
b) x = 0 or x = 3
15
c) −2a
94.
4
m
d)
16n4
85.
b) x = 0.5
95.
a) 27n
n+1 n+4
b) 2
3
86. (a) 32
(b)
87.
31
1
3
(c) 4
(d)
1
125
(e)
1
16
96.
c) x ∈ R
3
d) x ∈ ∅
a) 0
e) x ∈ (−∞, 0)
b) 0
97.
c) −4.5
c) x ∈ [− 74 , ∞)
e) 2 log10 a
f) 5a − 2
98. Graph omitted. Solution: x ∈ (8, ∞)
(a) 12
99. Graph omitted. Solution: x ∈ (0, 16).
(b) 32
100.
Examinable Section
90.
Non-Examinable Section
101. Omitted.
1
c)
ln a − 2 ln b − 5 ln c
2
102. Omitted.
a) 2x
Chapter 5
b) 3y − 1
Pre-tutorial exercises
c) 7
92.
a) x ∈ (−3, 22]
b) x ∈ (0, 14 ]
a) 4 log3 x + log3 (x + 3)
b) 3 log8 x − log8 (x − 8)
91.
a) x ∈ (−∞, 41 )
b) x ∈ ( 54 , ∞)
d) 23
89.
a) x ∈ (−∞, 3]
b) x ∈ [−2, ∞)
a) m 15
7
a) x = 1.54 (2 d.p.)
b) x = −0.82 (2 d.p.)
b) 32 a 15 b− 5
88.
a) x = 5 or x = −2
103.
(a) No
d) mp
(b) No
e) 5x
(c) No
a) x = −3
(d) Yes
b) x = 2
Examinable Section
c) x = −2
104.
d) x = −
(a) dom(f ) = R \ {−4}
(b) dom(f ) = (−∞, 2]
1
5
(c) dom(f ) = R \ {±1}
93.
22
(d) dom(f ) = [−1, 1]
(c)
(e) dom(f ) = (−1, 1)
(f) dom(f ) = R
105.
• (f − g)(x) = x2 − 2x + 1 −
dom(f − g) = [0, ∞)
(a) dom(f ) = ( 12 , ∞)
(b) dom(f ) = (−∞, 1)
5
(a) dom(f ) = R \ {0}
(b)
dom(f ) = R \ { 12 }
(d) dom(f ) = R
(d)
(e) dom(f ) = [0, ∞)
(g) dom(f ) = (−∞, −1] ∪ [1, ∞)
(h) dom(f ) = − 12 , ∞
(j) dom(f ) = (−∞, 1)
(k) dom(f ) = R \ {1, 3}
x
1
(f + g)(x) = x−4
+ 2x+3
3
dom(f + g) = R \ {4, − 2 }
(e)
f
g
√
2
• (f − g)(x) = log3 x −
dom(f − g) = (0, 1]
x
• (f g)(x) = (x−4)(2x+3)
dom(f g) = R \ {4, − 32 }
• fg (x) = x(2x+3)
x−4
dom fg = R \ {4, − 32 }
• (f + g)(x) = log2 (x + 2) + x21−1
dom(f + g) = (−2, ∞) \ {±1}
2
(x) = 9−x
2x
f
dom g = [−3, 0) ∪ (0, 3]
√
• (f + g)(x) = log3 x + 1 − x
dom(f + g) = (0, 1]
•
x
1
• (f − g)(x) = x−4
− 2x+3
dom(f − g) = R \ {4, − 32 }
(b)
x
√
• (f g)(x) = 2x 9 − x2
dom(f g) = [−3, 3]
(i) dom(f ) = (0, ∞)
•
f
g
√
√
• (f − g)(x) = 9 − x2 − 2x
dom(f − g) = [−3, 3]
(f) dom(f ) = (−∞, 0]
(a)
√
√
√
(x) = x −2x+1
x
f
dom g = (0, ∞)
√
• (f + g)(x) = 9 − x2 + 2x
dom(f + g) = [−3, 3]
•
(c) dom(f ) = R
107.
3
• (f g)(x) = x 2 − 2x 2 +
dom(f g) = [0, ∞)
(c) dom(f ) = (1, ∞)
106.
• (f + g)(x) = x2 − 2x + 1 +
dom(f + g) = [0, ∞)
√
1−x
√
• (f g)(x) = (log3 x) 1 − x
dom(f g) = (0, 1]
•
log3 x
(x) = √
1−x
dom fg = (0, 1)
f
g
108. dom(f ) = [−5, 2] ∪ [3, ∞)
• (f − g)(x) = log2 (x + 2) − x21−1
dom(f − g) = (−2, ∞) \ {±1}
109.
• (f g)(x) = logx22(x+2)
−1
dom(f g) = (−2, ∞) \ {±1}
• fg (x) = (x2 − 1) log2 (x + 2)
dom fg = (−2, ∞) \ {±1}
110.
(a) f (g(x)) = (2x + 4)2
(b) g(f (x)) = 2x2 + 4
(a) [−1, ∞)
(b) [1, ∞)
(c) [1, ∞)
(d) [−1, ∞)
(e) f −1 (x) = x2 − 2x
23
x
x
(f)
119.
y
(a) Graph omitted
f −1 (x) = − √1x − 2
dom(f −1 ) = (0, ∞)
ran(f −1 ) = (−∞, −2)
(3, 3)
y = f (x)
2
(b) Graph omitted
g −1 (x) = √1x − 2
dom(g −1 ) = (0, ∞)
ran(g −1 ) = (−2, ∞)
(−1, 1)
(1, −1)
2
x
y = f −1 (x)
111.
√
120. f −1 (x) = − 4 − x
121.
(a) dom(f ◦ g) = (−∞, −2)
(b) f (g(x)) = |x|
112.
(a) f ◦ g(x) = |x|
(b) dom(f ◦ g) = (−∞, −1] ∪ [1, ∞)
4x2 − 2 if x > 12
113. f (g(x)) =
1 − 2x
if x ≤ 12
114.
a) g ◦ f : [ 12 , 10] → R, g(f (x)) = 6x + 2
c) g ◦ g : [4, 35
3 ] → R, g(g(x)) = 9x − 28
d) f ◦ f : (−4, 72 ] → R, f (f (x)) = 4x + 9
115. dom(f ) = R \ { 95 }
9x − 3
f −1 (x) =
5x − 2
ran(f −1 ) = R \ { 95 }
116. (b), (c) and (d)
a) Omitted
b) dom(f ) = [−2, 2] and ran(f ) = [0, 2]
c) Since f is not one–one, f does not have
an inverse function.
118.
(a) b = 0, f −1 (x) =
(b) b = −2, f
−1
(b) dom(f ) = (−∞, −1]
[−1, ∞)
ran(f ) =
(c) dom(f −1 ) = [−1, ∞)
(−∞, −1]
√
(d) f −1 (x) = −1 − x + 1
ran(f −1 ) =
122. S = [−1, ∞)
b) f ◦ g : [4, 17
3 ] → R, f (g(x)) = 6x − 11
117.
(a) b = −1
√
x+4
√
(x) = − x − 2
24