RATIONAL FUNCTIONS, EQUATIONS, AND INEQUALITIES 11

1. 10.6.2
0

2. •It is an expression that
can be written as a ratio
of two polynomials.
•It can be described as a
function where either the
numerator, denominator,
or both have a variable
2
𝑥
𝑥2
+ 2𝑥 + 3
𝑥 + 1
5
𝑥 − 3

3. IDENTIFY THE FOLLOWING ALGEBRAIC
EXPRESSION IF THEY ARE RATIONAL OR
NOT
1
3𝑥3
𝑥 + 1
𝑥3 − 1
𝑥−2
− 5
𝑥3 − 1

4. AN EQUATION INVOLVING
RATIONAL EXPRESSIONS
EXAMPL
E 5
𝑥
−
3
2𝑥
=
1
5

5. AN INEQUALITY INVOLVING
RATIONAL EXPRESSIONS
EXAMPL
E 5
𝑥 − 3
≤
2
𝑥
<
≤
≥
>

6. •A function of the
form 𝑓 𝑥 =
𝑃(𝑥)
𝑄(𝑥)
where p(x) and q(x)
are polynomial
functions and q(x)
is not equal to zero
function q(x) ≠ 0.
EXAMPL
E𝑓 𝑥 = 𝑦
𝑓 𝑥 =
𝑥2
+ 2𝑥 + 3
𝑥 + 1

7. There are the restrictions on the 𝒙 −
𝒗𝒂𝒍𝒖𝒆𝒔 of a reduced rational function.
To find the restrictions equate the
denominator to 0 to solve or 𝒙.

8. Let 𝒏 be the degree of numerator and 𝒎 be the degree of
denominator.
• If 𝒏 < 𝒎, 𝒚 = 𝟎
• If 𝒏 = 𝒎, 𝒚 =
𝒂
𝒃
, where a is the leading coefficient of the
numerator and b is the leading coefficient of the
denominator.
•If 𝒏 > 𝒎, there is no horizontal asymptotes.

9. • 𝟓𝒙
• 𝒙 − 𝟒
• 𝟐𝒙 𝟑 − 𝒙 − 𝟒
•8
• 𝒙 𝟐 − 𝟐𝒙 𝟓 − 𝒙
• 𝒚 𝟐 − 𝒚 + 𝟏
• 𝟏
•1
•3
•0
•5
•2

10. To find the
vertical
asymptote:
The vertical
asymptote is 𝒙 = 𝟓.
To find the
horizontal
asymptote:
The horizontal
asymptote is 𝒚 = 𝟎.

11. To find the vertical
asymptote:
The graph has the
line 𝒙 = −𝟐 as
vertical asymptote.
To find the horizontal
asymptote:
The graph has the line
𝒚 = 𝟒 as a horizontal

12. To find the vertical
asymptote:
The vertical
asymptotes
are
𝒙 = −
𝟏
𝟐
and

13. To find the horizontal
asymptote:
The graph has the line
𝒚 = 𝟎 as a horizontal
asymptote.

14. To find the vertical
asymptote:
The vertical
asymptotes
are
𝒙 = −
𝟓
𝟑
and

15. To find the horizontal
asymptote:
The graph has
𝒏𝒐 horizontal asymptote.

16. To find the vertical
asymptote:
The graph has the
line
𝒙 = 𝟒 as vertical
asymptote.

17. To find the horizontal
asymptote:
The graph has the
line
𝒚 = 𝟏 as horizontal
asymptote.

18. 10.7.20

19. •The domain is a rational
function 𝒇 𝒙 =
𝑵(𝒙)
𝑫(𝒙)
is all the
values of 𝒙 that will not make
𝑫 𝒙 equal to zero.

20. •To find the range of rational function
is by finding the domain of the
inverse function.
•Another way is to find the range of
rational function is to find the value
of horizontal asymptote.

21. 𝒇 𝒙 =
𝟐
𝒙 − 𝟑
𝒙 − 𝟑 = 𝟎
𝒙 = 𝟑
The domain of
𝒇(𝒙) is the set
of all real
numbers
except 3.
𝒚 =
𝟐
𝒙 − 𝟑
𝒙 =
𝟐
𝒚 − 𝟑
𝒙(𝒚 − 𝟑) = 𝟐
𝒙𝒚 − 𝟑𝒙 = 𝟐
𝒙𝒚 = 𝟐 + 𝟑𝒙
𝒚 =
𝟐 + 𝟑𝒙
𝒙
𝒙 = 𝟎
The range of
𝒇(𝒙) is the set
of all real
numbers
except 0.

22. 𝒇 𝒙 =
𝒙 − 𝟓
𝒙 + 𝟐
𝒙 + 𝟐 = 𝟎
𝒙 = −𝟐
The domain of
𝒇(𝒙) is the set
of all real
numbers
except -2.
𝒚 =
𝒙 − 𝟓
𝒙 + 𝟐
𝒙 =
𝒚 − 𝟓
𝒚 + 𝟐
𝒙 𝒚 + 𝟐 = 𝒚 − 𝟓
𝒙𝒚 + 𝟐𝒙 = 𝒚 − 𝟓
𝒙𝒚 − 𝒚 = −𝟓 − 𝟐𝒙
𝒚 =
−𝟓 − 𝟐𝒙
𝒙 − 𝟏
𝒙 = 𝟏
The range of
𝒇(𝒙) is the set
of all real
numbers
except 1.
𝒚(𝒙 − 𝟏) = −𝟓 − 𝟐𝒙

23. 𝒇 𝒙 =
(𝒙 − 𝟒)(𝒙 + 𝟐)
(𝒙 − 𝟑)(𝒙 − 𝟏)
𝒙 − 𝟑 = 𝟎 𝒙 − 𝟏 = 𝟎
𝒙 = 𝟑 𝒙 = 𝟏
The domain of 𝒇(𝒙) is
the set of all real
numbers except 3
and 1.

24. 𝒚 =
𝒂
𝒃
=
𝟏
𝟏
= 𝟏
The range of 𝒇(𝒙) is
the set of all real
numbers except 1.
𝒇(𝒙) =
𝒙 𝟐
− 𝟔𝒙 − 𝟖
𝒙 𝟐 − 𝟒𝒙 + 𝟑

25. 𝒇 𝒙 =
𝟑𝒙 − 𝟗
𝒙 𝟐 − 𝒙 − 𝟔
𝒙 − 𝟑 = 𝟎 𝒙 + 𝟐 = 𝟎
𝒙 = 𝟑 𝒙 = −𝟐
The domain of 𝒇(𝒙) is
the set of all real
numbers except 3
and -2.
𝒇 𝒙 =
𝟑𝒙 − 𝟗
(𝒙 − 𝟑)(𝒙 + 𝟐)

26. 𝒚 = 𝟎
The range of 𝒇(𝒙) is
the set of all real
numbers except 0.
𝒇(𝒙) =
𝟑𝒙 − 𝟗
𝒙 𝟐 − 𝒙 − 𝟔

27. 𝒚 = 𝟎
The range of 𝒇(𝒙) is
the set of all real
numbers except 0.
𝒇(𝒙) =
𝟑𝒙 − 𝟗
𝒙 𝟐 − 𝒙 − 𝟔

28. 10.9.20

30. EXAMPLE
:
Solve for
𝑥:
2
𝑥
−
3
2𝑥
=
1
5

31. EXAMPLE
:
Solve for
𝑥:
𝑥 + 3
𝑥 − 1
=
4
𝑥 − 1

33. EXAMPLE
:
Solve for
𝑥:
𝑥
𝑥 + 2
−
1
𝑥 − 2
=
8
𝑥2 − 4

36. 10.12/13.20

37. Interval Set
Notation
Graph
(a, b) {𝐱⃓ 𝐚 < 𝐱⃓ < 𝐛}
[a, b] {𝐱⃓ 𝐚 ≤ 𝐱⃓ ≤ 𝐛}
[a, b) {𝐱⃓ 𝐚 ≤ 𝐱⃓ < 𝐛}

38. Interval Set
Notation
Graph
(a, ∞) {𝒙 𝒂 < 𝒙}
[a, ∞) {𝒙 𝒂 ≤ 𝒙}
(−∞, b) {𝒙 𝒙 < 𝒃}
(−∞, b] {𝒙 𝒙 ≤ 𝒃}

39. EXAMPLE
: Solve the inequality
2𝑥
𝑥+1
≥ 1.

40. EXAMPLE
: Solve the inequality
2𝑥
𝑥+1
≥ 1.

41. EXAMPLE
: Solve the inequality
2𝑥
𝑥+1
≥ 1.
−1 1
𝑥 < −1 −1 < 𝑥 < 1 𝑥 > 1

42. EXAMPLE
: Solve the inequality
2𝑥
𝑥+1
≥ 1.
INTERVAL 𝒙 < −𝟏 −𝟏 < 𝒙 < 𝟏 𝒙 > 𝟏
TEST POINT 𝒙 = −𝟐 𝒙 = 𝟎 𝒙 = 𝟐
𝒙 − 𝟏
𝒙 + 𝟏
𝒙 − 𝟏
𝒙 + 𝟏
−
−
−
−+
+
+
+
+

43. EXAMPLE
: Solve the inequality
2𝑥
𝑥+1
≥ 1.
−1 1
𝑥 < −1 −1 < 𝑥 < 1 𝑥 > 1

44. EXAMPLE
: Solve the inequality
3
𝑥−2
<
1
𝑥
.

45. EXAMPLE
: Solve the inequality
3
𝑥−2
<
1
𝑥
.

46. EXAMPLE
: Solve the inequality
3
𝑥−2
<
1
𝑥
.
−1 2
𝑥 < −1 −1 < 𝑥 < 0 𝑥 > 2
0
0 < 𝑥 < 2

47. EXAMPLE
: Solve the inequality
3
𝑥−2
<
1
𝑥
.
INTERVAL 𝒙 < −𝟏 −𝟏 < 𝒙 < 𝟎 𝟎 < 𝒙 < 𝟐 𝒙 > 𝟐
TEST
POINT
𝒙 = −𝟐 𝒙 = −
𝟏
𝟐
𝒙 = 𝟏 𝒙 = 𝟑
𝟐(𝒙 + 𝟏)
𝒙
𝒙 − 𝟐
𝟐(𝒙 + 𝟏)
− +
+
++
+
+
+
−
−
−
−−
−
− +

48. EXAMPLE
: Solve the inequality
3
𝑥−2
<
1
𝑥
.
−1 2
𝑥 < −1 −1 < 𝑥 < 0 𝑥 > 2
0
0 < 𝑥 < 2

49. EXAMPLE
: Solve the inequality
𝑥+12
𝑥+2
≥ 2.

50. EXAMPLE
: Solve the inequality
𝑥+12
𝑥+2
≥ 2.

51. EXAMPLE
: Solve the inequality
𝑥+12
𝑥+2
≥ 2.
8
𝐴
−2
𝐵 𝐶

52. EXAMPLE
: Solve the inequality
𝑥+12
𝑥+2
≥ 2.
𝐴 = −3

53. EXAMPLE
: Solve the inequality
𝑥+12
𝑥+2
≥ 2.
𝐵 = 0
.

54. EXAMPLE
: Solve the inequality
𝑥+12
𝑥+2
≥ 2.
𝐶 = 9

55. EXAMPLE
: Solve the inequality
𝑥+12
𝑥+2
≥ 2.
8
𝐴
−2
𝐵 𝐶

56. 10.16.2
0

59. EXAMPLE
:

60. EXAMPLE
:

61. EXAMPLE
:

62. EXAMPLE
:

63. EXAMPLE
:

64. 10.19.2
0

65. EXAMPLE
:

66. EXAMPLE
:

67. EXAMPLE
:

68. EXAMPLE
:

69. EXAMPLE
:

70. EXAMPLE
:

71. 10.20.2
0

72. RECALL:
 The DOMAIN of a function is the set of all values
that the variable x can take.
 The RANGE of a function is the set of all values that
f(x) will take.
 The ZEROS of a function are the values of x which
make the function zero.
 INTERCEPTS are x- or y- intercepts crosses the x-
axis or y-axis.
 Y - INTERCEPTS is the y-coordinate of the point
where the graph crosses the y-axis
 X - INTERCEPTS is the x-coordinate of the point
where the graph crosses the x-axis

73. There are the restrictions on the 𝒙 −
𝒗𝒂𝒍𝒖𝒆𝒔 of a reduced rational function.
To find the restrictions equate the
denominator to 0 to solve or 𝒙.

74. Let 𝒏 be the degree of numerator and 𝒎 be the degree of
denominator.
• If 𝒏 < 𝒎, 𝒚 = 𝟎
• If 𝒏 = 𝒎, 𝒚 =
𝒂
𝒃
, where a is the leading coefficient of the
numerator and b is the leading coefficient of the
denominator.
•If 𝒏 > 𝒎, there is no horizontal asymptotes.

76. EXAMPLE
:
𝒙 = 𝟎
𝟎 = 𝟎 =
There is
no
x-
𝒚 =
There is
no
y-
intercept.
𝒙 = 𝟎
𝒚 = 𝟎
𝒏 < 𝒎

77. EXAMPLE
:
x -4 -3 -2 -1 0 1 2 3 4
y -
1.7
5
-
2.3
3
-3.5 -7 und 7 3.5 2.3
3
1.7
5

78. EXAMPLE
:
𝟐𝒙 − 𝟖 = 𝟎
𝟎 =
𝟓𝒙 = 𝟎
𝒚 =
𝒚 = 𝟎
𝒚 =
𝒏 = 𝒎
𝟐𝒙 = 𝟖
𝒙 = 𝟒
𝒙 = 𝟎
𝟎 =
𝒙 = 𝟎
𝒙 = 𝟒 𝒚 =

79. 𝒙 = 𝟒 𝒚 = (𝟎, 𝟎) (𝟎, 𝟎)
𝒙 = 𝟎INTERV
AL
𝒙 < 𝟎 𝟎 < 𝒙 < 𝟑 𝒙 > 𝟒
TEST
POINT
𝒙 = −𝟏 𝒙 = −𝟏 𝒙 = −𝟏
𝟓𝒙
𝟐𝒙 − 𝟖
𝟓𝒙
𝟐𝒙 − 𝟖
−
−
−
−
+ +
+
++

80. EXAMPLE
:
𝒙 + 𝟐 = 𝟎
𝟎 =
𝒙 − 𝟐 = 𝟎
𝒚 =
𝒚 = −𝟏
𝒚 =
𝒏 = 𝒎
𝒙 = −𝟐
𝒙 = 𝟐
𝟎 =
𝒙 = 𝟐
𝒙 = −𝟐 𝒚 =

81. 𝒙 = −𝟐 𝒚 = (𝟐, 𝟎) (𝟎, −𝟏)
𝒙 = 𝟐INTERV
AL
𝒙 < −𝟐 −𝟐 < 𝒙 < 𝟐 𝒙 > 𝟐
TEST
POINT
𝒙 = −𝟑 𝒙 = 𝟎 𝒙 = 𝟑
𝒙 + 𝟐
𝒙 − 𝟐
𝒙 + 𝟐
𝒙 − 𝟐
−
−
−
−
+ +
+
++

82. 10.22.2
0

85. EXAMPLE
:

86. EXAMPLE
:

87. EXAMPLE
:

89. 10.23.2
0

90. x -2 -1 0 1 2
y -5 -3 -1 1 3
x -5 -3 -1 1 3
y -2 -1 0 1 2

92. 3 steps to find the inverse of a one-to-one
function;

93. EXAMPLE
:

94. EXAMPLE
:

95. EXAMPLE
:

96. EXAMPLE
:

97. EXAMPLE
:

98. EXAMPLE
:

99. EXAMPLE
:

100. EXAMPLE
: