2. Relations & Functions
A relation is a set of any ordered pairs that are related in any way.
e.g. x 2 y 2 25
3. Relations & Functions
A relation is a set of any ordered pairs that are related in any way.
e.g. x 2 y 2 25
A function is a relation such that for any x value, there is a maximum
of one y value.
e.g. y x 2
4. Relations & Functions
A relation is a set of any ordered pairs that are related in any way.
e.g. x 2 y 2 25
A function is a relation such that for any x value, there is a maximum
of one y value.
e.g. y x 2
Straight Line Test
If a straight line is drawn parallel to the y axis, it will only cross a
function once, if at all.
5. Relations & Functions
A relation is a set of any ordered pairs that are related in any way.
e.g. x 2 y 2 25
A function is a relation such that for any x value, there is a maximum
of one y value.
e.g. y x 2
Straight Line Test
If a straight line is drawn parallel to the y axis, it will only cross a
function once, if at all.
y 1
y
x
x
6. Relations & Functions
A relation is a set of any ordered pairs that are related in any way.
e.g. x 2 y 2 25
A function is a relation such that for any x value, there is a maximum
of one y value.
e.g. y x 2
Straight Line Test
If a straight line is drawn parallel to the y axis, it will only cross a
function once, if at all.
y 1
y
x
x
7. Relations & Functions
A relation is a set of any ordered pairs that are related in any way.
e.g. x 2 y 2 25
A function is a relation such that for any x value, there is a maximum
of one y value.
e.g. y x 2
Straight Line Test
If a straight line is drawn parallel to the y axis, it will only cross a
function once, if at all.
y 1
y
x
x
function
8. Relations & Functions
A relation is a set of any ordered pairs that are related in any way.
e.g. x 2 y 2 25
A function is a relation such that for any x value, there is a maximum
of one y value.
e.g. y x 2
Straight Line Test
If a straight line is drawn parallel to the y axis, it will only cross a
function once, if at all.
y 1 y x y2
y
x
x x
function
9. Relations & Functions
A relation is a set of any ordered pairs that are related in any way.
e.g. x 2 y 2 25
A function is a relation such that for any x value, there is a maximum
of one y value.
e.g. y x 2
Straight Line Test
If a straight line is drawn parallel to the y axis, it will only cross a
function once, if at all.
y 1 y x y2
y
x
x x
function
10. Relations & Functions
A relation is a set of any ordered pairs that are related in any way.
e.g. x 2 y 2 25
A function is a relation such that for any x value, there is a maximum
of one y value.
e.g. y x 2
Straight Line Test
If a straight line is drawn parallel to the y axis, it will only cross a
function once, if at all.
y 1 y x y2
y
x
function
x x
function
11. Relations & Functions
A relation is a set of any ordered pairs that are related in any way.
e.g. x 2 y 2 25
A function is a relation such that for any x value, there is a maximum
of one y value.
e.g. y x 2
Straight Line Test
If a straight line is drawn parallel to the y axis, it will only cross a
function once, if at all.
y 1 y x y2
y
x
function
x x
note: actually two functions
function
y x and y x
13. Domain and Range y f x
Domain: All possible values of x that can be substituted into the
function/relation.
14. Domain and Range y f x
Domain: All possible values of x that can be substituted into the
function/relation.
“Domain is the INPUT of the function/relation”
15. Domain and Range y f x
Domain: All possible values of x that can be substituted into the
function/relation.
“Domain is the INPUT of the function/relation”
To find a domain, look for values x could not be.
16. Domain and Range y f x
Domain: All possible values of x that can be substituted into the
function/relation.
“Domain is the INPUT of the function/relation”
To find a domain, look for values x could not be.
e.g.
y x y2
x
17. Domain and Range y f x
Domain: All possible values of x that can be substituted into the
function/relation.
“Domain is the INPUT of the function/relation”
To find a domain, look for values x could not be.
e.g.
y x y2
x
domain: x 0
18. Domain and Range y f x
Domain: All possible values of x that can be substituted into the
function/relation.
“Domain is the INPUT of the function/relation”
To find a domain, look for values x could not be.
e.g.
y x y2 y y f x
3
1
x 2 x
domain: x 0
19. Domain and Range y f x
Domain: All possible values of x that can be substituted into the
function/relation.
“Domain is the INPUT of the function/relation”
To find a domain, look for values x could not be.
e.g.
y x y2 y y f x
3
1
x 2 x
domain: x 0 domain: x 0 and x 2
21. Things to look for:
1. Fractions: bottom of fraction 0
22. Things to look for:
1. Fractions: bottom of fraction 0
1
e.g. i y
x
23. Things to look for:
1. Fractions: bottom of fraction 0
1
e.g. i y
x
x0
24. Things to look for:
1. Fractions: bottom of fraction 0
1
e.g. i y
x
x0
domain: all real x except x 0
25. Things to look for:
1. Fractions: bottom of fraction 0
1 1
e.g. i y ii y
x x2 1
x0
domain: all real x except x 0
26. Things to look for:
1. Fractions: bottom of fraction 0
1 1
e.g. i y ii y
x x2 1
x0 x2 1 0
domain: all real x except x 0
x2 1
x 1
27. Things to look for:
1. Fractions: bottom of fraction 0
1 1
e.g. i y ii y
x x2 1
x0 x2 1 0
domain: all real x except x 0
x2 1
x 1
domain: all real x except x 1
28. Things to look for:
1. Fractions: bottom of fraction 0
1 1
e.g. i y ii y
x x2 1
x0 x2 1 0
domain: all real x except x 0
x2 1
x 1
domain: all real x except x 1
4x 3
iii y
x 1 7 x
29. Things to look for:
1. Fractions: bottom of fraction 0
1 1
e.g. i y ii y
x x2 1
x0 x2 1 0
domain: all real x except x 0
x2 1
x 1
domain: all real x except x 1
4x 3
iii y
x 1 7 x
x 1 0
x 1
30. Things to look for:
1. Fractions: bottom of fraction 0
1 1
e.g. i y ii y
x x2 1
x0 x2 1 0
domain: all real x except x 0
x2 1
x 1
domain: all real x except x 1
4x 3
iii y
x 1 7 x
x 1 0 7x 0
x 1 x7
31. Things to look for:
1. Fractions: bottom of fraction 0
1 1
e.g. i y ii y
x x2 1
x0 x2 1 0
domain: all real x except x 0
x2 1
x 1
domain: all real x except x 1
4x 3
iii y
x 1 7 x
x 1 0 7x 0
x 1 x7
domain: all real x except x 1 or 7
33. 2. Root Signs: you can’t find the square root of a negative number.
34. 2. Root Signs: you can’t find the square root of a negative number.
e.g. i y 4 x 2
35. 2. Root Signs: you can’t find the square root of a negative number.
e.g. i y 4 x 2
4 x2 0
x2 4
36. 2. Root Signs: you can’t find the square root of a negative number.
e.g. i y 4 x 2
4 x2 0
x2 4
domain: 2 x 2
37. 2. Root Signs: you can’t find the square root of a negative number.
e.g. i y 4 x 2 ii y x 3 5 x
4 x2 0
x2 4
domain: 2 x 2
38. 2. Root Signs: you can’t find the square root of a negative number.
e.g. i y 4 x 2 ii y x 3 5 x
4 x2 0 x3 0
x2 4 x 3
domain: 2 x 2
39. 2. Root Signs: you can’t find the square root of a negative number.
e.g. i y 4 x 2 ii y x 3 5 x
4 x2 0 x3 0 5 x 0
x2 4 x 3 x5
domain: 2 x 2
40. 2. Root Signs: you can’t find the square root of a negative number.
e.g. i y 4 x 2 ii y x 3 5 x
4 x2 0 x3 0 5 x 0
x2 4 x 3 x5
domain: 2 x 2 domain: 3 x 5
41. 2. Root Signs: you can’t find the square root of a negative number.
e.g. i y 4 x 2 ii y x 3 5 x
4 x2 0 x3 0 5 x 0
x2 4 x 3 x5
domain: 2 x 2 domain: 3 x 5
1
iii y
x2
42. 2. Root Signs: you can’t find the square root of a negative number.
e.g. i y 4 x 2 ii y x 3 5 x
4 x2 0 x3 0 5 x 0
x2 4 x 3 x5
domain: 2 x 2 domain: 3 x 5
1
iii y
x2
x20
43. 2. Root Signs: you can’t find the square root of a negative number.
e.g. i y 4 x 2 ii y x 3 5 x
4 x2 0 x3 0 5 x 0
x2 4 x 3 x5
domain: 2 x 2 domain: 3 x 5
1
iii y
x2
x20
domain: x 2
45. Range: All possible y values obtained by substituting in the domain
“Range is the OUTPUT of the function/relation”
46. Range: All possible y values obtained by substituting in the domain
“Range is the OUTPUT of the function/relation”
e.g.
y x y2
x
47. Range: All possible y values obtained by substituting in the domain
“Range is the OUTPUT of the function/relation”
e.g.
y x y2
x
range: all real y
48. Range: All possible y values obtained by substituting in the domain
“Range is the OUTPUT of the function/relation”
e.g.
y x y2 y y f x
3
1
x 2 x
range: all real y
49. Range: All possible y values obtained by substituting in the domain
“Range is the OUTPUT of the function/relation”
e.g.
y x y2 y y f x
3
1
x 2 x
range: all real y range: y 1 and y 3
51. Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always 0
52. Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always 0
e.g. i y x 2
53. Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always 0
e.g. i y x 2
range: y 0
54. Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always 0
e.g. i y x 2 ii y x 2 3
range: y 0
55. Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always 0
e.g. i y x 2 ii y x 2 3
range: y 0 y 03
56. Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always 0
e.g. i y x 2 ii y x 2 3
range: y 0 y 03
range: y 3
57. Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always 0
e.g. i y x 2 ii y x 2 3
range: y 0 y 03
range: y 3
iii y 5 x 2
58. Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always 0
e.g. i y x 2 ii y x 2 3
range: y 0 y 03
range: y 3
iii y 5 x 2
y 50
59. Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always 0
e.g. i y x 2 ii y x 2 3
range: y 0 y 03
range: y 3
iii y 5 x 2
y 50
range: y 5
60. Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always 0
e.g. i y x 2 ii y x 2 3
range: y 0 y 03
range: y 3
iii y 5 x 2 iv y x 2
y 50
range: y 5
61. Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always 0
e.g. i y x 2 ii y x 2 3
range: y 0 y 03
range: y 3
iii y 5 x 2 iv y x 2
y 50 range: y 0
range: y 5
62. Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always 0
e.g. i y x 2 ii y x 2 3
range: y 0 y 03
range: y 3
iii y 5 x 2 iv y x 2
y 50 range: y 0
range: y 5
v y x 2 5
63. Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always 0
e.g. i y x 2 ii y x 2 3
range: y 0 y 03
range: y 3
iii y 5 x 2 iv y x 2
y 50 range: y 0
range: y 5
v y x 2 5
y 05
64. Things to look for:
1. Maximum/Minimum values: even powers and absolute values
are always 0
e.g. i y x 2 ii y x 2 3
range: y 0 y 03
range: y 3
iii y 5 x 2 iv y x 2
y 50 range: y 0
range: y 5
v y x 2 5
y 05
range: y 5
67. 2. Restrictions on Domain: sub in endpoints and centre of domain
e.g. y 4 x 2
68. 2. Restrictions on Domain: sub in endpoints and centre of domain
e.g. y 4 x 2
domain: 2 x 2
69. 2. Restrictions on Domain: sub in endpoints and centre of domain
e.g. y 4 x 2 when x 2, y 4 22
domain: 2 x 2 0
70. 2. Restrictions on Domain: sub in endpoints and centre of domain
e.g. y 4 x 2
when x 2, y 4 22 when x 0, y 4 02
domain: 2 x 2 0 2
71. 2. Restrictions on Domain: sub in endpoints and centre of domain
when x 2, y 4 22 when x 0, y 4 0
2
e.g. y 4 x 2
domain: 2 x 2 0 2
range: 0 y 2
72. 2. Restrictions on Domain: sub in endpoints and centre of domain
when x 2, y 4 22 when x 0, y 4 0
2
e.g. y 4 x 2
domain: 2 x 2 0 2
range: 0 y 2
3. Fractions:
73. 2. Restrictions on Domain: sub in endpoints and centre of domain
when x 2, y 4 22 when x 0, y 4 0
2
e.g. y 4 x 2
domain: 2 x 2 0 2
range: 0 y 2
3. Fractions: If you have a constant on the top of the fraction, fraction 0
74. 2. Restrictions on Domain: sub in endpoints and centre of domain
when x 2, y 4 22 when x 0, y 4 0
2
e.g. y 4 x 2
domain: 2 x 2 0 2
range: 0 y 2
3. Fractions: If you have a constant on the top of the fraction, fraction 0
1
e.g. i y
x
75. 2. Restrictions on Domain: sub in endpoints and centre of domain
when x 2, y 4 22 when x 0, y 4 0
2
e.g. y 4 x 2
domain: 2 x 2 0 2
range: 0 y 2
3. Fractions: If you have a constant on the top of the fraction, fraction 0
1
e.g. i y
x
y0
76. 2. Restrictions on Domain: sub in endpoints and centre of domain
when x 2, y 4 22 when x 0, y 4 0
2
e.g. y 4 x 2
domain: 2 x 2 0 2
range: 0 y 2
3. Fractions: If you have a constant on the top of the fraction, fraction 0
1
e.g. i y
x
y0
range: all real y except y 0
77. 2. Restrictions on Domain: sub in endpoints and centre of domain
when x 2, y 4 22 when x 0, y 4 0
2
e.g. y 4 x 2
domain: 2 x 2 0 2
range: 0 y 2
3. Fractions: If you have a constant on the top of the fraction, fraction 0
1
e.g. i y
1 ii y 5
x x
y0
range: all real y except y 0
78. 2. Restrictions on Domain: sub in endpoints and centre of domain
when x 2, y 4 22 when x 0, y 4 0
2
e.g. y 4 x 2
domain: 2 x 2 0 2
range: 0 y 2
3. Fractions: If you have a constant on the top of the fraction, fraction 0
1
e.g. i y
1 ii y 5
x x
y0 y 50
range: all real y except y 0
79. 2. Restrictions on Domain: sub in endpoints and centre of domain
when x 2, y 4 22 when x 0, y 4 0
2
e.g. y 4 x 2
domain: 2 x 2 0 2
range: 0 y 2
3. Fractions: If you have a constant on the top of the fraction, fraction 0
1
e.g. i y
1 ii y 5
x x
y0 y 50
range: all real y except y 0 range: all real y except y 5
80. 2. Restrictions on Domain: sub in endpoints and centre of domain
when x 2, y 4 22 when x 0, y 4 0
2
e.g. y 4 x 2
domain: 2 x 2 0 2
range: 0 y 2
3. Fractions: If you have a constant on the top of the fraction, fraction 0
1
e.g. i y
1 ii y 5
x x
y0 y 50
range: all real y except y 0 range: all real y except y 5
x7
iii y
x4
81. 2. Restrictions on Domain: sub in endpoints and centre of domain
when x 2, y 4 22 when x 0, y 4 0
2
e.g. y 4 x 2
domain: 2 x 2 0 2
range: 0 y 2
3. Fractions: If you have a constant on the top of the fraction, fraction 0
1
e.g. i y
1 ii y 5
x x
y0 y 50
range: all real y except y 0 range: all real y except y 5
x7 1
iii y x4 x7
x4
x4
3
82. 2. Restrictions on Domain: sub in endpoints and centre of domain
e.g. y 4 x 2
when x 2, y 4 22 when x 0, y 4 02
domain: 2 x 2 0 2
range: 0 y 2
3. Fractions: If you have a constant on the top of the fraction, fraction 0
1
e.g. i y
1 ii y 5
x x
y0 y 50
range: all real y except y 0 range: all real y except y 5
x7 1
iii y x4 x7
x4
3 x4
y 1 3
x4
83. 2. Restrictions on Domain: sub in endpoints and centre of domain
e.g. y 4 x 2
when x 2, y 4 22 when x 0, y 4 02
domain: 2 x 2 0 2
range: 0 y 2
3. Fractions: If you have a constant on the top of the fraction, fraction 0
1
e.g. i y
1 ii y 5
x x
y0 y 50
range: all real y except y 0 range: all real y except y 5
x7 1
iii y x4 x7
x4
3 x4
y 1 3
x4
y 1 0
84. 2. Restrictions on Domain: sub in endpoints and centre of domain
e.g. y 4 x 2
when x 2, y 4 22 when x 0, y 4 02
domain: 2 x 2 0 2
range: 0 y 2
3. Fractions: If you have a constant on the top of the fraction, fraction 0
1
e.g. i y
1 ii y 5
x x
y0 y 50
range: all real y except y 0 range: all real y except y 5
x7 1
iii y x4 x7
x4
3 x4
y 1 3
x4
y 1 0
range: all real y except y 1
91. Function Notation
e.g. f x 3 x 2 4
a) f 5 3 5 4 b) f a 3a 2 4
2
75 4
79
92. Function Notation
e.g. f x 3 x 2 4
a) f 5 3 5 4 b) f a 3a 2 4
2
75 4
79
c) f x h f x
93. Function Notation
e.g. f x 3 x 2 4
a) f 5 3 5 4 b) f a 3a 2 4
2
75 4
79
c) f x h f x 3 x h 4 3x 2 4
2
94. Function Notation
e.g. f x 3 x 2 4
a) f 5 3 5 4 b) f a 3a 2 4
2
75 4
79
c) f x h f x 3 x h 4 3x 2 4
2
3 x 2 6 xh 3h 2 4 3 x 2 4
6 xh 3h 2
95. Function Notation
e.g. f x 3 x 2 4
a) f 5 3 5 4 b) f a 3a 2 4
2
75 4
79
c) f x h f x 3 x h 4 3x 2 4
2
3 x 2 6 xh 3h 2 4 3 x 2 4
6 xh 3h 2
Exercise 2F; 1, 2, 3acdfi, 4begh, 5a, 6, 7a, 8abd,
10abdf, 11aceh, 12bd, 14*