1. Relations & Functions

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 

12. Domain and Range y  f  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

20. Things to look for:
1. Fractions:

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
x0

24. Things to look for:
1. Fractions: bottom of fraction  0
1
e.g.  i  y 
x
x0
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
x0
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
x0 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
x0 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
x0 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
x0 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
x0 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 7x  0
x 1 x7

31. Things to look for:
1. Fractions: bottom of fraction  0
1 1
e.g.  i  y   ii  y 
x x2 1
x0 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 7x  0
x 1 x7
domain: all real x except x  1 or 7

32. 2. Root Signs:

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 x3 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 x3 0 5 x  0
x2  4 x  3 x5
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 x3 0 5 x  0
x2  4 x  3 x5
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 x3 0 5 x  0
x2  4 x  3 x5
domain:  2  x  2 domain:  3  x  5
1
 iii  y 
x2

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 x3 0 5 x  0
x2  4 x  3 x5
domain:  2  x  2 domain:  3  x  5
1
 iii  y 
x2
x20

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 x3 0 5 x  0
x2  4 x  3 x5
domain:  2  x  2 domain:  3  x  5
1
 iii  y 
x2
x20
domain: x  2

44. Range: All possible y values obtained by substituting in the domain

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

50. Things to look for:
1. Maximum/Minimum values:

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  03

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  03
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  03
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  03
range: y  3
 iii  y  5  x 2
y  50

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  03
range: y  3
 iii  y  5  x 2
y  50
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  03
range: y  3
 iii  y  5  x 2  iv  y  x  2
y  50
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  03
range: y  3
 iii  y  5  x 2  iv  y  x  2
y  50 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  03
range: y  3
 iii  y  5  x 2  iv  y  x  2
y  50 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  03
range: y  3
 iii  y  5  x 2  iv  y  x  2
y  50 range: y  0
range: y  5
v y  x  2  5
y  05

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  03
range: y  3
 iii  y  5  x 2  iv  y  x  2
y  50 range: y  0
range: y  5
v y  x  2  5
y  05
range: y  5

65. 2. Restrictions on Domain:

66. 2. Restrictions on Domain: sub in endpoints and centre of domain

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
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

72. 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:

73. 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

74. 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 
x

75. 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 
x
y0

76. 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 
x
y0
range: all real y except y  0

77. 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
y0
range: all real y except y  0

78. 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
y0 y  50
range: all real y except y  0

79. 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
y0 y  50
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
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
y0 y  50
range: all real y except y  0 range: all real y except y  5
x7
 iii  y 
x4

81. 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
y0 y  50
range: all real y except y  0 range: all real y except y  5
x7 1
 iii  y  x4 x7
x4
x4
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
y0 y  50
range: all real y except y  0 range: all real y except y  5
x7 1
 iii  y  x4 x7
x4
3 x4
y  1 3
x4

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
y0 y  50
range: all real y except y  0 range: all real y except y  5
x7 1
 iii  y  x4 x7
x4
3 x4
y  1 3
x4
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
y0 y  50
range: all real y except y  0 range: all real y except y  5
x7 1
 iii  y  x4 x7
x4
3 x4
y  1 3
x4
y  1 0
range: all real y except y  1

85. Function Notation

86. Function Notation
e.g. f  x   3 x 2  4

87. Function Notation
e.g. f  x   3 x 2  4
a) f  5

88. Function Notation
e.g. f  x   3 x 2  4
a) f  5  3  5  4
2

89. Function Notation
e.g. f  x   3 x 2  4
a) f  5  3  5  4
2
 75  4
 79

90. Function Notation
e.g. f  x   3 x 2  4
a) f  5  3  5  4 b) f  a 
2
 75  4
 79

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*