This document discusses different types of functions including polynomials, rational functions, radical functions, absolute value functions, exponential functions, logarithmic functions, and trigonometric functions. It provides examples of how to sketch graphs of various functions by completing the square, reflecting, shifting, compressing/expanding, and using properties of exponentials, logarithms, and trigonometric functions. Key aspects like periodicity and domains/ranges are also covered.

1. Types of a Function
Chapter 2
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2. Types of
Functions
Polynomials
Radical
Functions
Rational
Functions
Absolute Value
Functions
Exponential
Functions
Logarithmic
Functions
Trigonometric
Functions
Hyperbolic
Functions
2

3. Polynomials
Linear Quadratic Cubic
3

4. • A polynomial of degree 1 is of the form 𝑓 𝑥 = 𝑚𝑥 + 𝑏 and so it is a linear
function.
• A polynomial of degree 2 is of the form 𝒇 𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 and is called a
quadratic function.
• The parabola opens upward if 𝑎 > 0 and downward if 𝑎 < 0 .
• A polynomial of degree 3 is of the form 𝒇 𝒙 = 𝒂𝒙𝟑 + 𝒃𝒙𝟐 + 𝒄𝒙 + 𝒅 and is
called a cubic function.
4
slope Y-intercept

5. Completing Square
𝑥 ±
𝑎
2
2
−
𝑎
2
2
± 𝑏
5

6. Example 2.1
6
Sketch 𝑓 𝑥 = 𝑥2
+ 6𝑥 + 10
Completing the square, we write the equation of the graph as
y = 𝑥2
+ 6𝑥 + 10 = 𝑥 + 3 2
+ 1
This means we
obtain the
desired graph by
starting with the
parabola and
shifting 3 units to
the left and then
1 unit upward.
Base Graph

7. Rational Functions
Square Root Cubic Root
7

8. Sketch the Following Functions in Steps
8
𝑓 𝑥 = 𝑥 − 1 − 2
𝑓 𝑥 = 2 − 1 − 𝑥
𝑓 𝑥 = − −𝑥
Example 2.2

9. Example 2.2.a
9
Sketch 𝑓 𝑥 = 𝑥 − 1 − 2

10. Example 2.2.b
10
Sketch 𝑓 𝑥 = 2 − 1 − 𝑥

11. Example 2.2.c
11
Sketch 𝑓 𝑥 = − −𝑥

12. Some Special Radical Functions
12

13. Sketch the Following Functions in Steps
Example 2.3
𝑓 𝑥 =
1
𝑥 − 1
+ 3
𝑓 𝑥 =
1
(𝑥 − 1)2
+ 4
𝑓 𝑥 =
1
2(1 − 𝑥)3
+ 5

14. Example 2.3.a
14
Graph 𝑓 𝑥 =
1
𝑥−1
+ 3

15. Example 2.3.b
15
Graph 𝑓 𝑥 =
1
(𝑥−1)2 + 4

16. Example 2.3.c
16
Graph 𝑓 𝑥 =
1
2(1−𝑥)3 + 5
Challenging
Problem

17. Absolute Value Functions
17
𝑎 = 𝑎 𝑖𝑓 𝑎 ≥ 0
𝑎 = −𝑎 𝑖𝑓 𝑎 < 0
(Remember that if a is
negative, then "a is
positive.)
𝑥 = ቊ
−𝑥, 𝑥 < 0
𝑥, 𝑥 ≥ 0

18. Example 2.4
18
Sketch the Following Functions in Steps
𝑓 𝑥 = 𝑥 − 2 + 1
𝑓 𝑥 = 𝑥2
− 1

19. Example 2.4.a
19
Graph 𝑓 𝑥 = 𝑥2
− 1 in steps
We first graph the parabola
𝑓 𝑥 = 𝑥2
− 1 by shifting the parabola
𝑦 = 𝑥2 downward 1 unit. We see that
the graph lies below the x-axis when
− 1 < 𝑥 < 1, so we reflect that part of
the graph about the x-axis to obtain the
graph of 𝑓 𝑥 = 𝑥2
− 1

20. Example 2.4.b
20
Graph 𝑓 𝑥 = 𝑥 − 2 + 1 in steps

21. Try to sketch in steps and then find domain &
Range
𝑦 = (𝑥 + 1)2
𝑦 = 𝑥2
𝑦 = 𝑥2
+ 3
𝑦 = (𝑥 + 1)2
+3
𝑦 = 𝑥2
+ 2𝑥 + 1
𝑦 = 𝑥2
− 4𝑥 + 7
𝑦 = 2𝑥2
− 8𝑥 + 14
𝑦 =
2
𝑥 − 3
𝑦 =
2𝑥 − 3
𝑥 − 1
𝑦 = 𝑥 − 1 − 1
𝑦 =
3
𝑥 − 1 − 1
𝑦 = 1 − 𝑥2
𝑦 = 12 − 4𝑥 − 𝑥2 − 2
𝑦 =
2𝑥 − 3
𝑥 − 1
𝑦 = 𝑥 − 1 − 1

22. Trigonometric Functions
Sine Cosine Tangent
22

23. • An important property of the sine and cosine functions is that they are
periodic functions and have period 2𝜋.
• The period of tangent is 𝜋.
• The remaining three trigonometric functions (cosecant, secant, and cotangent)
are the reciprocals of the sine, cosine, and tangent functions; therefore, they
have the same period.
23
Notice that for both the sine and cosine functions the domain is and the range
is the closed interval . Thus, for all values of , we have
−1 ≤ 𝑠𝑖𝑛𝑥 ≤ 1 − 1 ≤ 𝑐𝑜𝑠𝑥 ≤ 1
or, in terms of absolute values, 𝑠𝑖𝑛𝑥 ≤ 1 𝑐𝑜𝑠𝑥 ≤ 1

24. Example 2.5 Sketch the Following Functions
𝑓(𝑥) = 1 + 4𝑐𝑜𝑠3𝑥
𝑓 𝑥 = 𝑠𝑖𝑛2𝑥
𝑓 𝑥 = 1 − 𝑠𝑖𝑛𝑥
𝑓(𝑥) =
1
4
𝑡𝑎𝑛 𝑥 −
𝜋
4

25. Example 2.5.a
25
Sketch the Graph 𝑓 𝑥 = 𝑠𝑖𝑛2𝑥
We obtain the graph of 𝑓 𝑥 from
that of 𝑓 𝑥 = 𝑠𝑖𝑛𝑥 by
compressing horizontally by a
factor of 2. Thus, whereas the
period of 𝑓 𝑥 = 𝑠𝑖𝑛𝑥 is 2𝜋, the
period of 𝑓 𝑥 = 𝑠𝑖𝑛2𝑥 is
2𝜋
2
= π .

26. Example 2.5.b
26
Sketch the Graph 𝑓 𝑥 = 1 − 𝑠𝑖𝑛𝑥
To obtain the graph of 𝑓 𝑥 = 1 −
𝑠𝑖𝑛𝑥, we again start with 𝑓 𝑥 =
𝑠𝑖𝑛𝑥 . We reflect about the x-axis
to get the graph of 𝑓 𝑥 = −𝑠𝑖𝑛𝑥
and then we shift 1 unit upward to
get 𝑓 𝑥 = 1 − 𝑠𝑖𝑛𝑥.

27. Example 2.5.c
27
Sketch The graph 𝑓(𝑥) =
1
4
𝑡𝑎𝑛 𝑥 −
𝜋
4

28. Example 2.5.d Sketch The Graph 𝑓(𝑥) = 1 + 4𝑐𝑜𝑠3𝑥

29. Trigonometric Functions
Cosecant Secant Cotangent
29

30. Basic Trigonometric Identities
30
𝑠𝑖𝑛2
𝜃 + 𝑐𝑜𝑠2
𝜃 = 1
𝑡𝑎𝑛2
𝜃 + 1 = 𝑠𝑒𝑐2
𝜃
1 + 𝑐𝑜𝑡2
𝜃 = 𝑐𝑠𝑐2
𝜃
sin −𝜃 = −𝑠𝑖𝑛𝜃
cos −𝜃 = 𝑐𝑜𝑠𝜃
cos 𝐴 + 𝐵 = 𝑐𝑜𝑠𝐴𝑐𝑜𝑠𝐵 − 𝑠𝑖𝑛𝐴𝑠𝑖𝑛𝐵
sin 𝐴 + 𝐵 = 𝑠𝑖𝑛𝐴𝑐𝑜𝑠𝐵 + 𝑐𝑜𝑠𝐴𝑠𝑖𝑛𝐵
𝑐𝑜𝑠2𝜃 = 𝑐𝑜𝑠2
𝜃 − 𝑠𝑖𝑛2
𝜃 = 2𝑐𝑜𝑠2
𝜃 − 1
𝑠𝑖𝑛2𝜃 = 2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃
𝑐𝑜𝑠2
𝜃 =
1 + 𝑐𝑜𝑠2𝜃
2
𝑠𝑖𝑛2
𝜃 =
1 − 𝑐𝑜𝑠2𝜃
2

31. 31
tan2
𝜃 + 1 = sec2
𝑥
We start with the identity 𝑠𝑖𝑛2
𝜃 + 𝑐𝑜𝑠2
𝜃 = 1
Dividing both sides of this equation by 𝑐𝑜𝑠2
𝜃,
we obtain
𝑠𝑖𝑛2𝜃
𝑐𝑜𝑠2𝜃
+ 1 =
1
𝑐𝑜𝑠2𝜃
Since
𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃
= 𝑡𝑎𝑛𝜃 and
1
𝑐𝑜𝑠𝜃
= 𝑠𝑒𝑐𝜃, we
conclude that tan2
𝜃 + 1 = sec2
𝑥

32. Exponential Functions
32
𝑒𝑥+𝑦
= 𝑒𝑥
𝑒𝑦
𝑒𝑥−𝑦
=
𝑒𝑥
𝑒𝑦
(𝑒𝑥
)𝑦
= 𝑒𝑥𝑦
(𝑎𝑏)𝑥
= 𝑎𝑥
𝑏𝑦
𝑒−𝑥
=
1
𝑒𝑥

33. Example 2.6
33
𝑓 𝑥 = 𝑒𝑥+1
33
𝑓 𝑥 = 3 − 2𝑥
Sketch the Following Graph in Steps
𝑓 𝑥 =
1
2
𝑒−𝑥
− 1
𝑓 𝑥 = −(3𝑥
+ 2)
𝑓 𝑥 = 2𝑥
+ 1
𝑓 𝑥 = −3−𝑥

34. Example 2.6.a
34
Sketch the Graph 𝑓 𝑥 = 3 − 2𝑥
First, we reflect the graph
of 𝑦 = 2𝑥
about the x-axis
to get the graph of 𝑦 =
− 2𝑥
Base
graph
Then, we shift the
graph of 𝑦 = −2𝑥
upward 3 units to
obtain the graph of
𝑦 = 3 − 2𝑥
The Domain is ℝ
The Range is (-∞,3)

35. Example 2.6.b
35
Sketch the Graph 𝑓 𝑥 =
1
2
𝑒−𝑥
− 1
Base
graph
First, we reflect the graph
of 𝑦 = 𝑒𝑥 about the y-axis
to get the graph of 𝑦 =
𝑒−𝑥
. (Notice that the graph
crosses the y-axis with a
slope of -1)
Then, Then we
compress the graph
vertically by a factor
of 2 to obtain the
graph of 𝑦 =
1
2
𝑒−𝑥
Finally, we shift the
graph downward one
unit to get 𝑓 𝑥 =
1
2
𝑒−𝑥 − 1
The Domain is ℝ
The Range is (-1, ∞)

36. Example 2.6.c
36
Sketch the Graph 𝑓 x = e𝑥+1
36

37. Example 2.6.d
37
Sketch the Graph 𝑓 x = −(3𝑥
+ 2)
37

38. Example 2.6.e
38
Sketch the Graph 𝑓 x = 2𝑥
+ 1
38

39. Example 2.6.f
39
Sketch the Graph 𝑓 x = −3−𝑥
39

40. Logarithmic Functions
40
log𝑎(𝑎𝑥) = 𝑥 𝑥 = ℝ
ln 𝑒𝑥 = 𝑥
𝑙𝑛𝑥 = 𝑦 ⇔ 𝑒𝑦
= 𝑥
log𝑎(𝑥𝑟) = rlog𝑎 𝑥
log𝑎(𝑥𝑦) = log𝑎 𝑥 + log𝑎 𝑦 log𝑎
𝑥
𝑦
= log𝑎 𝑥 − log𝑎 𝑦
𝑎log𝑎 𝑥 = 𝑥 𝑥 > 0
log𝑎(𝑥) =
𝑙𝑛𝑥
𝑙𝑛𝑎
for any positive numbers
𝑙𝑛𝑒 = 1
𝑒𝑙𝑛𝑥 = 𝑥 𝑥 > 0

41. Example 2.7
41
Simplify the Following Expressions
41
𝑒ln(7.2)
𝑙𝑛
sin2 𝑥 tan4 𝑥
𝑥2 + 1 2
𝑙𝑛
4 𝑥2 + 1
𝑥2 − 1
𝑙𝑛𝑒(2 ln 𝑥)
ln[ln 𝑒𝑒 ]
2ln( 𝑒)
𝑒x−ln(𝑥)
𝑒ln(𝑥2+𝑦2)
𝑒 ln 𝑥 −ln 𝑦
𝑒−ln(𝑥2)
ln 𝑒𝑒𝑥
𝑙𝑛
𝑥3 + 1 4sin2 𝑥
3
𝑥
ln(𝑠𝑖𝑛𝜃) − ln
𝑠𝑖𝑛𝜃
5
ln(3𝑥2 − 9𝑥) + ln
1
3𝑥
ln(𝑠𝑒𝑐𝜃) + ln(𝑐𝑜𝑠𝜃)
3𝑙𝑛
3
𝑡2 − 1 − ln(𝑡 + 1)
𝑒5−3𝑥
= 10

42. Example 2.8
42
Solve the Equation 𝑒5−3𝑥
= 10
42
𝑙𝑛𝑒5−3𝑥
= 𝑙𝑛10
5 − 3𝑥 = 𝑙𝑛10
3𝑥 = 5 − 𝑙𝑛10
𝑥 =
1
3
5 − 𝑙𝑛10
We take natural logarithms
of both sides of the equation
Apply ln 𝑒𝑥 = 𝑥

43. 𝑒−ln(𝑥2)
𝑒
ln
1
𝑥2 =
1
𝑥2
43
Example 2.8

44. 𝑒ln(7.2)
7.2
44
Example 2.8

45. 𝑒 ln 𝑥 −ln 𝑦
𝑒 ln 𝑥 −ln 𝑦
= ൝
𝑒𝑙𝑛𝑥−𝑙𝑛𝑦
𝑥 ≥ 𝑦
𝑒𝑙𝑛𝑦−𝑙𝑛𝑥 𝑥 < 𝑦
= ቐ
ൗ
𝑥
𝑦 𝑥 ≥ 𝑦
ൗ
𝑦
𝑥 𝑥 < 𝑦
45
Example 2.8

46. 𝑒ln(𝑥2+𝑦2)
𝑥2 + 𝑦2
46
Example 2.8

47. 𝑒x−ln(𝑥)
𝑒𝑥
𝑒𝑙𝑛𝑥
=
𝑒𝑥
𝑥
47
Example 2.8

48. 2ln( 𝑒)
2 ln 𝑒
1
2 = 𝑙𝑛𝑒 = 1
48
Example 2.8

49. ln 𝑒𝑒𝑥
𝑒𝑥
49
Example 2.8

50. 𝑙𝑛𝑒(2 ln 𝑥)
ln 𝑒𝑙𝑛𝑥2
= 𝑙𝑛𝑥2
= 2𝑙𝑛𝑥
50
Example 2.8

51. 𝑙𝑛
4 𝑥2 + 1
𝑥2 − 1
𝑙𝑛
4 𝑥2 + 1
𝑥2 − 1
=
1
4
𝑙𝑛
𝑥2
+ 1
𝑥2 − 1
=
1
4
[ln 𝑥2
+ 1 − ln(𝑥2
− 1)
51
Example 2.8

52. 𝑙𝑛
sin2
𝑥 tan4
𝑥
𝑥2 + 1 2
ln sin2
𝑥 + ln tan4
𝑥 − ln 𝑥2
+ 1 2
= 2 ln 𝑠𝑖𝑛𝑥 + 4 ln 𝑡𝑎𝑛𝑥 − 2ln(𝑥2
+ 1)
52
Example 2.8

53. 𝑙𝑛
𝑥3
+ 1 4
sin2
𝑥
3
𝑥
𝑙𝑛
𝑥3
+ 1 4
sin2
𝑥
3
𝑥
= 4 ln 𝑥3
+ 1 + 2 ln 𝑠𝑖𝑛𝑥 −
𝑙𝑛𝑥
3
53
Example 2.8

54. ln(𝑠𝑖𝑛𝜃) − ln
𝑠𝑖𝑛𝜃
5
ln(𝑠𝑖𝑛𝜃) − ln
𝑠𝑖𝑛𝜃
5
= ln 𝑠𝑖𝑛𝜃 − ln 𝑠𝑖𝑛𝜃 − −𝑙𝑛5 = 𝑙𝑛5
54
Example 2.8

55. ln(3𝑥2
− 9𝑥) + ln
1
3𝑥
= 𝑙𝑛3𝑥 𝑥 − 3 + 𝑙𝑛
1
3𝑥
= 𝑙𝑛 3𝑥 + 𝑙𝑛 𝑥 − 3 − 𝑙𝑛 3𝑥
= 𝑙𝑛(𝑥 − 3)
55
Example 2.8

56. ln(𝑠𝑒𝑐𝜃) + ln(𝑐𝑜𝑠𝜃)
ln 𝑐𝑜𝑠𝜃 −1
+ ln 𝑐𝑜𝑠𝜃 = 0
56
Example 2.8

57. 3𝑙𝑛
3
𝑡2 − 1 − ln(𝑡 + 1)
ln 𝑡2
− 1 − ln 𝑡 + 1 = ln 𝑡 − 1 𝑡 + 1 − ln 𝑡 + 1 = ln(𝑡 − 1)
57
Example 2.8

58. ln[ln 𝑒𝑒
]
𝑙𝑛𝑒 = 1
58
Example 2.8

59. Example 2.9
59
Sketch the Following Functions
59
𝑦 = ln 𝑥 − 2 − 1 𝑓 𝑥 = ln(𝑥 + 2)
𝑓 𝑥 = 𝑙𝑛𝑥 + 2
𝑓 𝑥 = 1 + ln(−𝑥)
𝑓 𝑥 = 𝑙𝑛 𝑥
𝑓 𝑥 = 2 − log(2 − 𝑥)

60. Example 2.9
60
Sketch the Graph 𝑦 = ln 𝑥 − 2 − 1
60
We start with the
graph of 𝑦 = 𝑙𝑛𝑥
Using the transformations, we
shift it 2 units to the right to get
the graph of 𝑦 = ln(𝑥 − 2)
Then we shift it 1 unit
downward to get the
graph of 𝑦 = 𝑥 − 2 − 1

61. Example 2.9
61
Sketch the Graph 𝑓 𝑥 = ln(𝑥 + 2)
61

62. 62
Sketch the Graph 𝑓 𝑥 = 𝑙𝑛𝑥 + 2
62
Example 2.9

63. 63
Sketch the Graph 𝑓 𝑥 = 1 + ln(−𝑥)
63
Example 2.9

64. 64
Sketch the Graph 𝑓 𝑥 = 𝑙𝑛 𝑥
64
Example 2.9

65. 65
Sketch the Graph 𝑓 𝑥 = 2 − log(2 − 𝑥)
65
Example 2.9

66. Hyperbolic Functions
Hyperbolic Sine of
x
𝑠𝑖𝑛ℎ𝑥 =
𝑒𝑥
− 𝑒−𝑥
2
𝑠𝑖𝑛ℎ0 = 0
Hyperbolic Cosine
of x
𝑐𝑜𝑠ℎ𝑥 =
𝑒𝑥
+ 𝑒−𝑥
2
𝑐𝑜𝑠ℎ0 = 1
Hyperbolic of
Tangent of x
𝑡𝑎𝑛ℎ𝑥 =
𝑠𝑖𝑛ℎ𝑥
𝑐𝑜𝑠ℎ𝑥
=
𝑒𝑥 − 𝑒−𝑥
𝑒𝑥 + 𝑒−𝑥
−1 < 𝑡𝑎𝑛ℎ𝑥 < 1
66

67. Exercise 2.10
cosh 2𝑙𝑛𝑥
Simplify the Following Expressions
sinh 5𝑙𝑛𝑥 cosh(𝑙𝑛3)
sinh(𝑙𝑛2)
If 𝑠𝑖𝑛ℎ𝑥 =
3
4
, find the values of the remaining five hyperbolic functions.

68. sinh(𝑙𝑛2)
Using the definition of sinh function, we write
sinh 𝑙𝑛2 =
𝑒𝑙𝑛2 − 𝑒−𝑙𝑛2
2
=
2 −
1
2
2
=
3
4
68

69. sinh 5𝑙𝑛𝑥
Using the definition of sinh function, we write
sinh 5𝑙𝑛𝑥 =
𝑒5𝑙𝑛𝑥 − 𝑒−5𝑙𝑛𝑥
2
=
𝑒ln 𝑥5
− 𝑒𝑙𝑛𝑥−5
2
=
𝑥5 − 𝑥−5
2
69

70. cosh(𝑙𝑛3)
Using the definition of cosh function, we write
cosh 𝑙𝑛3 =
𝑒ln 3 + 𝑒−𝑙𝑛3
2
=
3 +
1
3
2
=
10
6
=
5
3
70

71. If 𝑠𝑖𝑛ℎ𝑥 =
3
4
, find the values of the remaining five
hyperbolic functions.
Using the identity cosh2 𝑥 − sinh2 𝑥 = 1, we see that
cosh2
𝑥 = 1 +
3
4
2
=
25
16
Since 𝑐𝑜𝑠ℎ𝑥 ≥ 1 for all x, we must have cosh 𝑥 =
5
4
.
Then, using the definitions for the other hyperbolic functions, we conclude
that 𝑡𝑎𝑛ℎ𝑥 =
3
5
, 𝑐𝑠𝑐ℎ𝑥 =
4
3
, 𝑠𝑒𝑐ℎ𝑥 =
4
5
, 𝑐𝑜𝑡ℎ𝑥 =
5
3
.
71

72. Hyperbolic Functions
Hyperbolic
cotangent of x
𝑐𝑜𝑡ℎ𝑥 =
𝑐𝑜𝑠ℎ𝑥
𝑠𝑖𝑛ℎ𝑥
=
𝑒𝑥
+ 𝑒−𝑥
𝑒𝑥 − 𝑒−𝑥
Hyperbolic
secant of x
𝑠𝑒𝑐ℎ𝑥 =
1
𝑐𝑜𝑠ℎ𝑥
=
2
𝑒𝑥 + 𝑒−𝑥
0 < 𝑠𝑒𝑐ℎ𝑥 ≤ 1
Hyperbolic of
cosecant of x
cscℎ𝑥 =
1
𝑠𝑖𝑛ℎ𝑥
=
2
𝑒𝑥−𝑒−𝑥
72

73. Exercise 2.11
cosh −𝑥 = 𝑐𝑜𝑠ℎ𝑥
Prove the Following Hyperbolic Functions
sinh −𝑥 = −𝑠𝑖𝑛ℎ𝑥
𝑐𝑜𝑠ℎ𝑥 + 𝑠𝑖𝑛ℎ𝑥 = 𝑒𝑥
𝑐𝑜𝑠ℎ𝑥 − 𝑠𝑖𝑛ℎ𝑥 = 𝑒−𝑥
coth2
𝑥 − 1 = csch2
𝑥
sinh 𝑥 ± 𝑦 = 𝑠𝑖𝑛ℎ𝑥𝑐𝑜𝑠ℎ𝑦 ± 𝑐𝑜𝑠ℎ𝑥𝑠𝑖𝑛ℎ𝑦
cosh 𝑥 ± 𝑦 = 𝑐𝑜𝑠ℎ𝑥𝑐𝑜𝑠ℎ𝑦 ± 𝑠𝑖𝑛ℎ𝑥𝑠𝑖𝑛ℎ𝑦
tanh 𝑙𝑛𝑥 =
𝑥2 − 1
𝑥2 + 1
1 + 𝑡𝑎𝑛ℎ𝑥
1 − 𝑡𝑎𝑛ℎ𝑥
= 𝑒2𝑥

74. Assignment
2.1
2.1.1. 𝑦 = 𝑥2
+ 6𝑥 − 8
2.1.2. 𝑦 = 7 + 2𝑥 − 𝑥2
2.1.3. 𝑦 = 2𝑥 + 5
2.1.4. 𝑦 = 2𝑥 + 5
2.1.5. 𝑦 = 2 − 𝑥 + 3
2.1.6. 𝑦 =
3
x+2
2.1.7. 𝑦 =
𝑥−4
2𝑥+3
2.1.8. 𝑦 = −2 − 4𝑥 − 𝑥2
2.1.9. 𝑦 = −𝑠𝑖𝑛2𝑥
2.1.10. 𝑦 =
1
2
1 + 𝑒𝑥
2.1.11. 𝑓 𝑥 =
1
𝑥+2
2.1.12. 𝑓 𝑥 = 5𝑥+1
+ 2
2.1.13. 𝑓 𝑥 = 3 + 𝑙𝑛𝑥
2.1.14. 𝑓 𝑥 = ln 𝑥 − 1
2.1.15. 𝑓 𝑥 = 𝑙𝑜𝑔𝑥 − 1
2.1.16. 𝑓 𝑥 = ln −𝑥
2.1.17. 𝑓 𝑥 = 𝑒𝑥
+ 2
2.1.18. 𝑓 𝑥 = 𝑒−𝑥
− 1
2.1.19. 𝑓 𝑥 = 4 cos 2𝑥 −
𝜋
2
2.1.20. 𝑓 𝑥 = −3sin(𝜋𝑥 + 2)
74
Graph each of the following functions
then from the graph find its the domain
and range.