CAL1-CH0-NEW-CAL1-CH0-NEWCAL1-CH0-NEW.pdf

math

1. Lecture Notes for Calculus 101
Chapter 0 : Before Calculus
Feras Awad
Philadelphia University

2. Functions
Idea and Definition
Definition 1
A functionfis a rule that associates with each input, a unique
(exactly ONE
✓✗
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 2 / 81

3. Functions
Idea and Definition
Example 1
Letf(x) =3x
2
−4x+2. Then
f(−1)=3(−1)
2
−4(−1) +2=3+4+2=9
NOTE:There are 4-ways to represent functions:
Table of values Words Graph Formula
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 3 / 81

4. Functions
Some Graphs
y=xy=x
2
y=x
3
y=
√
xy=
3
√
xy=
1
x
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 4 / 81

5. Functions
Vertical Line Test
A curve in thexy−plane is the graph of some functionfif and only
if no vertical line intersects the curve more than once
Example 2
Which of the following graphs is a function?
✓✗✗
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 5 / 81

6. Functions
Domain (and Range) of Functions
The set of all allowable inputs
is theDomainoff.
The set of all resulting
outputs is theRangeoff.
NOTE:
possible domain, called theNatural Domain.
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 6 / 81

7. Functions
Examples on Function Domain
Example 3
Find the natural domain of the functionf(x) =x
3
−2x
2
+1.
dom(f)=all real numbers= (−∞,∞)=R−∞∞-2-1012
NOTE:
P(x) =anx
n
+an−1x
n−1
+· · ·+a1x+a0
has domainR.
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 7 / 81

8. Functions
Examples on Function Domain
Example 4
Find the natural domain of the functionf(x) =
1
(x+1)(x−3)
.
dom(f)=all real numbers except−1 and 3=R− {−1,3}= (−∞,−1)∪(−1,3)∪(3,∞)−∞∞-13
NOTE:Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 8 / 81

9. Functions
Examples on Function Domain
Example 5
Find the natural domain of the functionf(x) =
√
x−1.
dom(f)=all real numbers such thatx−1≥0=allx∈Rsuch thatx≥1= [1,∞)−∞∞1
NOTE:
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 9 / 81

10. Functions
Examples on Function Domain
Example 6
Find the natural domain of the functionf(x) =
√
x
2
−5x+6.
dom(f)=all real numbers such thatx
2
−5x+6≥0(x−2)(x−3)≥0= (−∞,2]∪[3,∞)−∞∞23+−+
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 10 / 81

11. Functions
Examples on Function Domain
Example 7
Find the natural domain of the functionf(x) =
3
√
4−3x
2
.
dom(f)=all real numbers=RNOTE: +,−, 0 but depends on the function inside
it.
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 11 / 81

12. Functions
Examples on Function Domain
Example 8
Find the natural domain of the functionf(x) =
3
s
1
x
.
dom(f)=all real numbers except 0= (−∞,0)∪(0,∞)−∞∞0
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 12 / 81

13. Functions
Examples on Function Domain
Example 9
Find the natural domain of the functionf(x) =
√
x
x
2
−4
.
dom(f)=all real numbers such thatx≥0 except±2= [0,2)∪(2,∞)−∞∞-202
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 13 / 81

14. Functions
Examples on Function Domain
Example 10
Find the natural domain of the functionf(x) =
√
x+1−
√
2−x.
dom
“√
x+1
”
=x∈R;x+1≥0x≥ −1= [−1,∞)dom
“√
2−x
”
=x∈R;2−x≥0−x≥ −2x≤2= (−∞,2]−∞∞-1−∞∞2−∞∞-12∴dom(f) = [−1,∞)∩(−∞,2]= [−1,2]
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 14 / 81

15. Functions
Examples on Function Domain
Exercise 1
(1)Compare the domain of g(x) =x and f(x) =
x
2
+x
1+x
.
(2)Find the domain of each of the following.
a)f(x) =
p
x
2
+2x+4
b)g(x) =
x
x
2
−x−2
c)f(x) =
√
9−x
2
1−x
2
d)g(x) =
x−3
1−
√
x
e)f(x) =
x−3
1+
√
x
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 15 / 81

16. Functions
Piecewise Functions
Functions can be defined in pieces (cases).
Example 11
Letf(x) =





0 :x≤ −1
√
1−x
2
:−1<x<1
x :x≥1
f(0)=
√
1−0
2
=1f(−3)=0f(5)=5f(1)=1-111
NOTE:x=−1 andx=1 are called piecewise points.
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 16 / 81

17. Functions
Absolute Value
The absolute value ofx∈Ris defined by
|x|=
(
x:x≥0
−x:x<0
For example,|4|=4,
|0|=0,


−
1
2


=
1
2
.-111
NOTE:(1)|x|=a⇔x=±a.
Ex.|x|=4⇔x=±4
(2)|x| ≤a⇔ −a≤x≤a.
Ex.|x| ≤4⇔ −4≤x≤4
(3)|x| ≥a⇔x≥aorx≤ −a.
Ex.|x| ≥4⇔x≤ −4 orx≥4
(4)
√
x
2
=|x| (5) |g(x)|=dom(g(x))
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 17 / 81

18. Functions
Absolute Value
Example 12
Write the functiong(x) =|2x−1|as piecewise function.
g(x)=
(
2x−1:2x−1≥0
−(2x−1) :2x−1<0
=
(
2x−1:x≥
1/2
−2x+1:x<
1/2
g(x) =0⇒ |2x−1|=0⇒2x−1=0⇒x=
1/2−∞∞0.5+2x−1−1−2x
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 18 / 81

19. Functions
Trigonometric Ratios (Right Triangle)
acbθa:adjacentb:oppositec:hypotenusesinθ=
b
c
,cosθ=
a
c
,tanθ=
b
a
secθ=
c
a
,cscθ=
c
b
,cotθ=
a
b
NOTE:
secθ=
1
cosθ
,cscθ=
1
sinθ
,cotθ=
1
tanθ
tanθ=
sinθ
cosθ
,cotθ=
cosθ
sinθ
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 19 / 81

20. Functions
Angle Measures and Standard Position
We measure angles either by
degrees or by radians, where
πrad=180
◦
Example 13
90
◦
=90×
π
180
=
π
2
2π
3
=
2π
3
×
180
π
=120
◦
Initial RayVertexTerminal Rayθ
Counterclockwise
θis positive
Initial RayVertexTerminal Rayθ
Clockwise
θis negative
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 20 / 81

21. Functions
Angle Measures and Standard Position
Example 14
RightAcuteObtuseStraight90
◦
=
π/230
◦
=
π/6150
◦
=
5π/6180
◦
=π270
◦
=
3π/2−90
◦
=−
π/2−45
◦
=−
π/4−540
◦
=−3π
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 21 / 81

22. Functions
Some Trigonometric Values
θdeg0
◦
30
◦
45
◦
60
◦
90
◦
θrad0
π/6
π/4
π/3
π/2
sinθ
0
1/2
1/
√
2
√
3/21
cosθ
1
√
3/2
1/
√
2
1/20
tanθ
0
1/
√
31
√
3✗
cscθ
✗2
√
2
2/
√
31
secθ
1
2/
√
3
√
2 2 ✗
cotθ
✗
√
3 1
1/
√
30
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 22 / 81

23. Functions
Trigonometric Values in Coordinate Plane
-11-110
◦
90
◦
180
◦
270
◦
(x,y)1θxyQ1Q2Q3Q4All +sin+csc+tan+cot+cos+sec+
(x,y) = (cosθ,sinθ)
4 Quadrants
θ Pointsincos
0
◦
(1,0)0 1
90
◦
(0,1)1 0
180
◦
(−1,0)0−1
270
◦
(0,−1)−1 0
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 23 / 81

24. Functions
Trigonometric Functions
1
f(x) = sinx
Domain:R
Range:
h
−1,1
i
Period:2π
-11−2π−
3π
2
−π−
π
2
2π
3π
2
π
π
2
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 24 / 81

25. Functions
Trigonometric Functions
2
f(x) = cosx
Domain:R
Range:
h
−1,1
i
Period:2π
-11−2π−
3π
2
−π−
π
2
2π
3π
2
π
π
2
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 25 / 81

26. Functions
Trigonometric Functions
3
f(x) = tanx
Domain:x̸=±
π
2
,±
3π
2
,· · ·
Range:R
Period:π
-11−2π−
3π
2
−π−
π
2
2π
3π
2
π
π
2
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 26 / 81

27. Functions
Trigonometric Functions
4
f(x) = cotx
Domain:x̸=0,±π,±2π,· · ·
Range:R
Period:π
-11−2π−
3π
2
−π−
π
2
2π
3π
2
π
π
2
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 27 / 81

28. Functions
Trigonometric Functions
5
f(x) = secx
Domain:x̸=±
π
2
,±
3π
2
,· · ·
Range: (−∞,−1]∪[1,∞)
Period:2π
-11−2π−
3π
2
−π−
π
2
2π
3π
2
π
π
2
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 28 / 81

29. Functions
Trigonometric Functions
6
f(x) = cscx
Domain:x̸=0,±π,±2π,· · ·
Range: (−∞,−1]∪[1,∞)
Period:2π
-11−2π−
3π
2
−π−
π
2
2π
3π
2
π
π
2
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 29 / 81

30. Functions
Trigonometric Identities
sin
2
θ+ cos
2
θ=1
tan
2
θ+1= sec
2
θ
1+ cot
2
θ= csc
2
θ
sin(2θ) =2sinθcosθ
cos(2θ) = cos
2
θ−sin
2
θ
=2cos
2
θ−1
=1−2sin
2
θ
cos
2
θ=
1+ cos(2θ)
2
sin
2
θ=
1−cos(2θ)
2
sin(−θ) =−sinθ
cos(−θ) = cosθ
tan(−θ) =−tanθ
Ex:sin
„
−
π
6
«
=−sin
„
π
6
«
=−
1
2
tan
„
−
π
4
«
=−tan
„
π
4
«
=−1
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 30 / 81

31. Functions
Reference Angle
Example 15
Find all anglesθsuch that
cosθ=−
1/2.
θ1=π−
π
3
=
2π
3
±2nπθ2=π+
π
3
=
4π
3
±2nπwheren=0,1,2,· · ·0
◦
90
◦
180
◦
270
◦
απ−απ+α2π−α=
π/3✓✓
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 31 / 81

32. Functions
Reference Angle
Example 16
Find all anglesθsuch that
sinθ=−
1/
√
2.
θ1=π+
π
4
=
5π
4
±2nπθ2=2π−
π
4
=
7π
4
±2nπwheren=0,1,2,· · ·0
◦
90
◦
180
◦
270
◦
απ−απ+α2π−α=
π/4✓✓
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 32 / 81

33. Functions
Reference Angle
Example 17
Find all anglesθsuch that
tanθ=
√
3.
θ1=
π
3
±2nπθ2=π+
π
3
=
4π
3
±2nπ∴θ=
π
3
±nπwheren=0,1,2,· · ·0
◦
90
◦
180
◦
270
◦
απ−απ+α2π−α=
π/3✓✓
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 33 / 81

34. Functions
Reference Angle
Example 18
Find all anglesθsuch thatsinθ=0.
θ1=0±2nπθ2=π±2nπ∴θ=0±nπ=±nπwheren=0,1,2,· · ·
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 34 / 81

35. Functions
Reference Angle
Example 19
Find all anglesθsuch thatcosθ=0.
θ1=
π
2
±2nπθ2=
3π
2
±2nπ∴θ=
π
2
±nπwheren=0,1,2,· · ·
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 35 / 81

36. Functions
Reference Angle
Example 20
Find all anglesθsuch thatcosθ=−1.
θ=π±2nπ=
“
1±2n
”
πwheren=0,1,2,· · ·
Example 21 Find all anglesθsuch thatsinθ=1.
θ=
π
2
±2nπwheren=0,1,2,· · ·
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 36 / 81

37. Functions
Reference Angle
Example 22
Find the value ofsin
„
5π
3
«
.
2π−α=
5π
3
α=2π−
5π
3
=
π
3
∴sin
„
5π
3
«
=−sin
„
π
3
«
=−
√
3
2
0
◦
90
◦
180
◦
270
◦
απ−απ+α2π−α=
π/3✓
5π/3=300
◦
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 37 / 81

38. Functions
Reference Angle
Example 23
Find the value of
tan
“
−3π
4
”
=−tan
“
3π
4
”
π−α=
3π
4
α=π−
3π
4
=
π
4
∴tan
„
−3π
4
«
=−tan
„
3π
4
«
=−(−) tan
„
π
4
«
=10
◦
90
◦
180
◦
270
◦
απ−απ+α2π−α=
π/4✓
3π/4=135
◦
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 38 / 81

39. Functions
Reference Angle
Example 24
Find the domain off(x) = tanx
=
sinx
cosx
dom(tanx)=R−
n
cosx=0
o
=R−
n
x=
π/2±nπ
o
=dom(secx)
Example 25
Find the domain off(x) = cscx
=
1
sinx
dom(cscx)=R−
n
sinx=0
o
=R−
n
x=±nπ
o
=dom(cotx)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 39 / 81

40. New Functions from Old
Operations on Functions
Letfandgbe functions, then
„
f
×
+
−
g
«
(x)=f(x)
×
+
−
g(x) ;x∈dom(f)∩dom(g)(f÷g)(x)=f(x)÷g(x) ;x∈dom(f)∩dom(g)−
n
g(x) =0
o
Example 26
Letf(x) =
√
x
→dom(f) = [0,∞).
g(x) =
√
x
→dom(g) = [0,∞)
Find(f·g)(x)and its domain.
(f·g)(x)=
√
x·
√
x=
“√
x
”
2
=xdom(f·g)=[0,∞)∩[0,∞)= [0,∞)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 40 / 81

41. New Functions from Old
Composition of Functions
xgg(x)ff(g(x))f◦gf"circle"gofxf"after"gofx
Definition 2
(f◦g)(x) =f(g(x))
dom(f◦g) =set of allx∈dom(g)such thatg(x)∈dom(f)
NOTE: f◦g̸=g◦f.
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 41 / 81

42. New Functions from Old
Composition of Functions
Example 27
Given thatf(1) =1,f(−1) =0,g(−1) =1 andg(0) =0. Find
(f◦g)(−1).
(f◦g)(−1) =f(g(−1)) =f(1) =1Example 28
Letf(x) =x
2
+3 andg(x) =
√
x. then
(1)(f◦g)(x) =
f(g(x)) =f(
√
x) =
“√
x
”
2
+3=x+3dom(f◦g)=[0,∞)∩R= [0,∞)
(2)(g◦f)(x) =
g(f(x)) =g(x
2
+3) =
√
x
2
+3dom(g◦f)=R∩R=R
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 42 / 81

43. New Functions from Old
How Operations Affect Functions Graphs
(Translation)
Example
f(x) +cf(x)−cf(x+c)f(x−c)
f(x) =x
2
updownleftrightx
2
+cx
2
−c(x+c)
2
(x−c)
2
(0,0)(0,c)(0,−c)(−c,0)(c,0)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 43 / 81

44. New Functions from Old
How Operations Affect Functions Graphs
(Translation)
Example 29
Draw the graph off(x) = (x+1)
2
−1.
(0,0)(−1,0)(−1,−1)x
2
(x+1)
2
(x+1)
2
−1
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 44 / 81

45. New Functions from Old
How Operations Affect Functions Graphs
(Reflection)
Abouty−axisAboutx−axisf(−x)−f(x)f(x)f(−x)f(x)−f(x)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 45 / 81

46. New Functions from Old
How Operations Affect Functions Graphs
(Reflection)
Example 30
Draw the graph off(x) =4− |x−2|
(0,0)(2,0)(2,0)(2,4)|x||x−2|−|x−2|4− |x−2|
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 46 / 81

47. New Functions from Old
How Operations Affect Functions Graphs
(Reflection)
NOTE:
(1) f(x±c)
(2) f(−x)and/or−f(x)
(3) f(x)±c
Example 31
Draw the graph off(x) =2−
√
3−x
(0,0)(−3,0)(3,0)(3,0)(3,2)
√
x
√
x+3
√
−x+3−
√
3−x2−
√
3−x
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 47 / 81

48. New Functions from Old
Functions with Symmetric Graphs
Definition 3
f(x)is aneven functioniff(−x) =f(x). In this case,fis
symmetric about they−axis.−11
−3/2
3/2
−π/3
π/3x
2
|x|cosx
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 48 / 81

49. New Functions from Old
Functions with Symmetric Graphs
Definition 4
f(x)is anodd functioniff(−x) =−f(x). In this case,fis
symmetric about the origin.−11
−π/2
π/2
−π/4
π/4x
3
sinx−tanx= tan(−x)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 49 / 81

50. New Functions from Old
Functions with Symmetric Graphs
NOTE:
(1)
nor Even likef(x) =x+2cosx.
(2)▶Odd±Odd=Odd
▶Even±Even=Even
▶Odd×Odd=Even , Odd ÷Odd=Even
▶Even×Even=Even , Even÷Even=Even
▶Odd×Even=Odd , Odd ÷Even=Odd
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 50 / 81

51. Inverse Functions
Idea of Inverse Functions
When two functions "UNDO"
each other.The inverse function is denoted by
f
−1
and readf−inverse.
NOTE:
*f
−1
̸=
1
f
*y=x
3
, thenx=
3
√
y
* f, are the
outputs off
−1
and vice-versa.
xy=x
3
ff−inverse
domain
“
f
−1
”
=range(f)
range
“
f
−1
”
=domain(f)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 51 / 81

52. Inverse Functions
When Do the Inverse Exist?
Function
but NOT1−1
Function
and1−1A functionfhas inverse⇔it is one-to-one(1−1)⇔x1̸=x2,thenf(x1)̸=f(x2).⇔The graph intersects any horizontal⇔line at most once (Horizontal Line Test)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 52 / 81

53. Inverse Functions
When Do the Inverse Exist?
Example 32 Letf(x) =x
3
.
* fisR.* x1,x2∈Rsuch
thatx1̸=x2, then
x
3
1̸=x
3
2⇒f(x1)̸=f(x1)
* x
3
is 1−1 onR.
−11
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 53 / 81

54. Inverse Functions
When Do the Inverse Exist?
Example 33
Letf(x) =x
2
.
* fisR.* −1̸=1, but
(−1)
2
= (1)
2
.
* x
2
isNOT1−1 onR.* x
2
is 1−1 on[0,∞).
−11
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 54 / 81

55. Inverse Functions
Definition of Inverse Function
Definition 5
The functionsf,f
−1
are inverses provided both
“
f◦f
−1
”
(x) =f
“
f
−1
(x)
”
=x;x∈dom
“
f
−1
”
“
f
−1
◦f
”
(x) =f
−1
(f(x)) =x;x∈dom(f)
Example 34
Letf(x) =2x
3
+5x+3. Findxiff
−1
(x) =1.
f
−1
(x) =1⇒f
“
f
−1
(x)
”
=f(1)⇒x=f(1) =10
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 55 / 81

56. Inverse Functions
Finding the Inverse Function
To find the inverse function off(x):(1) y=f(x).
(2) xas a function ofy.(3) xbyf
−1
(x), andybyx.
Example 35
Findf
−1
(x)given thatf(x) =x
3
.
y=x
3
3
√
y=
3
√
x
3
x=
3
√
yf
−1
(x)=
3
√
xx
33
√
xy=x(3/4,27/64)(27/64,3/4)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 56 / 81

57. Inverse Functions
Finding the Inverse Function
Example 36
Findf
−1
(x)given that
f(x) =
√
1+2x.
y=
√
1+2x⇒y
2
=
“√
1+2x
”
2
⇒y
2
=1+2x⇒x=
y
2
−1
2
⇒f
−1
(x) =
1
2
“
x
2
−1
”√
1+2x
x
2
−1
2
y=x(1,0)(0,1)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 57 / 81

58. Inverse Functions
Finding the Inverse Function
Example 37
Findf
−1
(x)given thatf(x) =
x+1
x−1
.
y=
x+1
x−1
⇒(x−1)y=x+1⇒xy−y=x+1⇒xy−x=y+1⇒x(y−1) =y+1⇒x=
y+1
y−1
⇒f
−1
(x) =
x+1
x−1
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 58 / 81

59. Inverse Functions
Inverse Trigonometric Functions
f
−1
DomainRange Graph Note
sin
−1
x[−1,1]
h
−
π
2
,
π
2
i
−11π/2−π/2
arcsinx
cos
−1
x[−1,1][0, π]
−11π
arccosx
tan
−1
x R
“
−
π
2
,
π
2
”
π/2−π/2
arctanx
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 59 / 81

60. Inverse Functions
Inverse Trigonometric Functions
Example 38
Find the domain off(x) = sin
−1
(x−1).
−1≤x−1≤1⇒ −1
+1
≤x−1
+1
≤1
+1
⇒0≤x≤2⇒dom(f) = [0,2]
Example 39
Find the exact value.
(1)sin
−1
(
1/2)
=
π/6∈[
−π/2,
π/2]
(2)cos
−1
(
1/2)
=
π/3∈[0, π]
(3)tan
−1
(1)
=
π/4∈(
−π/2,
π/2)
(4)cos
−1
(2)
undefined
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 60 / 81

61. Inverse Functions
Inverse Trigonometric Functions
NOTE:
*sin
−1
(−x) =−sin
−1
x
*tan
−1
(−x) =−tan
−1
x*cos
−1
(−x) =π−cos
−1
x
*sin
−1
(x) + cos
−1
(x) =
π
2
*tan
−1
(x)̸=
sin
−1
(x)
cos
−1
(x)
Example 40
Find the exact value.
(1)sin
−1
(
−1/2)
=−sin
−1
(
1/2)=−
π/6∈[
−π/2,
π/2]
(2)tan
−1
(−1)
=−tan
−1
(1)=−
π/4∈(
−π/2,
π/2)
(3)cos
−1
(
−1/2)
=π−cos
−1
(
1/2)=π−
π/3=
2π/3∈[0, π]
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 61 / 81

62. Inverse Functions
Inverse Trigonometric Functions
Example 41
Find the exact value.
(1)sin
“
sin
−1
(
1/4)
”
=
1/4
Note that dom
“
sin
−1
”
= [−1,1]and
1/4∈[−1,1]Also,sinandsin
−1
are inverses and cancel each other
(2)tan (tan
−1
(
−17/9))
=
−17/9
Note that dom(tan
−1
) =Rand
−17/9∈RAlso,tanandtan
−1
are inverses and cancel each other
(3)cos (cos
−1
(
−2/3))
=
−2/3
Note that dom(cos
−1
) = [−1,1]and
−2/3∈[−1,1]Also,cosandcos
−1
are inverses and cancel each other
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 62 / 81

63. Inverse Functions
Inverse Trigonometric Functions
NOTE:
sin
−1
(sinx) =



π−x :
π/2≤x≤
3π/2
x−2nπ:x≥
3π/2
cos
−1
(cosx) =2nπ−x;ifx≥π
Example 42
Find the exact value.
(1)sin
−1
(sin (
5π/3))
= sin
−1
(sin (
5π/3−2π))= sin
−1
(sin (
−π/3))=
−π/3
(2)sin
−1
(sin (
4π/3))
= sin
−1
(sin (π−
4π/3))= sin
−1
(sin (
−π/3))=
−π/3
(3)cos
−1
(cos (
17π/4))
= cos
−1
(cos (4π−
17π/4))= cos
−1
(cos (
−π/4))= cos
−1
(cos (
π/4))=
π/4
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 63 / 81

64. Inverse Functions
Inverse Trigonometric Functions
Example 43
Find the exact value of
tan
−1
„
tan
„
5π
6
««
.
π−α=
5π
6
α=π−
5π
6
=
π
6
tan
−1
„
tan
„
5π
6
««
=−tan
−1
„
tan
„
π
6
««
=
−π
6
0
◦
90
◦
180
◦
270
◦
απ−απ+α2π−α
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 64 / 81

65. Inverse Functions
Inverse Trigonometric Functions
Example 44
Simplify the expressiontan
“
sin
−1
x
”
.
Let θ= sin
−1
x
sinθ= sin
“
sin
−1
x
”
sinθ=x;x∈[−1,1]∴tan
“
sin
−1
x
”
= tanθ=
x
√
1−x
2
wherex∈(−1,1)θx1
√
1−x
2
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 65 / 81

66. Inverse Functions
Inverse Trigonometric Functions
Example 45
Find the exact value ofsin (2sec
−1
3).
Let
θ= sec
−1
3
secθ= sec
“
sec
−1
3
”
secθ=3∴sin
“
2sec
−1
3
”
= sin(2θ) =2sinθcosθ=2×
√
8
3
×
1
3
=
4
√
2
9
θ
√
831
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 66 / 81

67. Exponential and Logarithmic Functions
The Exponential Function
Definition 6
The exponential function to the
basebisf(x) =b
x
whereb>0
andb̸=1.
domain ofb
x
=Rrange ofb
x
= (0,∞)b>12
x
0<b<1
“
1
2
”
x
(0,1)
Example 46
2
x
, π
x
,(
1/3)
x
,· · ·are exponentialx
2
,x
π
,x
1/3
,· · ·are NOT exponential
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 67 / 81

68. Exponential and Logarithmic Functions
The Exponential Function
Example 47
Find the domain of the functionf(x) =4
x
x
2
−4.
dom(f) =R− {x
2
−4=0}=R− {±2}NOTE:
The numbere≈2.7182818285· · ·is
called theNatural Number.
The exponential functionf(x) =e
x
is
called theNatural Exponential
Function.
e
x
(0,1)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 68 / 81

69. Exponential and Logarithmic Functions
The Exponential Function: Properties
(1)b
0
=1
Ex:2
0
=1
(2)b
x
·b
y
=b
x+y
Ex:2
4
·2
3
=2
7
(3)b
x
÷b
y
=b
x−y
Ex:
3
5
/3
2=3
3
(4)b
−x
=
1/b
x
Ex:e
−2
=
1/e
2
(5)(b
x
)
y
=b
xy
Ex:
“
3
2
”
3
=3
6
(6)
n
√
b
m
=b
m/n
Ex:
3
√
2
6
=2
6/3
=2
2
(7)(a·b)
x
=a
x
·b
x
Ex:(2e)
3
=2
3
·e
3
=8e
3
(8)(a÷b)
x
=a
x
÷b
x
Ex:(
5/2)
2
=
5
2
/2
2
(9)b
x
=b
y
⇔x=y
One-to-One Property
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 69 / 81

70. Exponential and Logarithmic Functions
The Exponential Function: Properties
Example 48
Find the exact value:
(1)(−8)
2
3
=
3
q
(−8)
2
=
3
√
64=4
(2)
−1/2
=
1
9
1/2
=
1
√
9
=
1
3Example 49
Solve the equation 2
x
=64.
2
x
=64⇒2
x
=2
6
⇒x=6
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 70 / 81

71. Exponential and Logarithmic Functions
The Logarithmic Function
Definition 7
The logarithmic function to the
basebisf(x) = log
bxwhere
b>0 andb̸=1.
domain oflog
bx= (0,∞)range oflog
bx=Rb>1log
2x0<b<1log1/2x(1,0)
NOTE: log
exis called theNatural Logarithmic
Functionand denoted bylnx.
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 71 / 81

72. Exponential and Logarithmic Functions
The Logarithmic Function
NOTE:
dom(log
bg(x)) =All real numbers such thatg(x)>0
and defined.
Example 50
Find the domain off(x) = ln (9−x
2
).
domln
“
9−x
2
”
=Allx∈Rsuch that 9−x
2
>0⇒x
2
<9⇒
√
x
2
<
√
9⇒ |x|<3⇒ −3<x<3⇒x∈(−3,3)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 72 / 81

73. Exponential and Logarithmic Functions
The Logarithmic Function: Properties
(1)log
b1=0
Ex:log
101=0
(2)log
bb=1
Ex:lne= log
ee=1
(3)log
b(x
n
) =nlog
bx
Ex:log
39= log
3
“
3
2
”
=2
(4)log
ba=
lna
lnb
Ex:log
43=
ln3/ln4
(5)log
b(xy) = log
bx+ log
by
(6)log
b(
x/y) = log
bx−log
by
(7)log
bx= log
by⇔x=y
One-to-One Property
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 73 / 81

74. Exponential and Logarithmic Functions
The Logarithmic Function: Properties
Example 51
Find the exact value.
(1)log
232
= log
2(2
5
)=5log
22=5
(2)log
42
= log
4
√
4= log
4
“
4
1/2
”
=
1/2
Example 52
Simplify the expressionlog
69−log
65+ log
620.
(log
69−log
65) + log
620= log
6
„
9
5
«
+ log
620= log
6
„
9
5
×20
«
= log
636= log
6
“
6
2
”
=2
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 74 / 81

75. Exponential and Logarithmic Functions
Logarithmic & Exponential are Inverses
FunctionDomainRange
b
x
R (0,∞)
log
bx (0,∞) R
log
b(b
x
)=x;x∈Rb
log
b
x
=x;x∈(0,∞)y=xb
x
log
bx
Example 53
Find the exact value of 5
2log
5
4
.
5
2log
5
4
=5
log
5(4
2
)
=4
2
=16
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 75 / 81

76. Exponential and Logarithmic Functions
Logarithmic & Exponential are Inverses
Example 54
Solve the following equations.
(1)e
x
=7
⇒ln (e
x
) = ln7⇒x= ln7
(2)log
10x=−2
⇒10
log
10
x
=10
−2
⇒x=
1
10
2
=0.01
(3)lnx=3
⇒e
lnx
=e
3
⇒x=e
3
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 76 / 81

77. Exponential and Logarithmic Functions
Logarithmic & Exponential are Inverses
Example 55
Solve the equatione
2x
+e
x
−6=0.
e
2x
+e
x
−6=0⇒(e
x
)
2
+e
x
−6=0(Lety=e
x
)⇒y
2
+y−6=0⇒(y+3)(y−2) =0Solution (1)⇒y=−3⇒e
x
=−3✗Solution (2)⇒y=2⇒e
x
=2 ✓⇒x= ln2
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 77 / 81

78. Exponential and Logarithmic Functions
Logarithmic & Exponential are Inverses
Example 56
Solve the equationln(x−2) + ln(2x−3) =2lnx.
ln
“
(x−2)(2x−3)
”
= ln
“
x
2
”
⇒(x−2)(2x−3) =x
2
⇒2x
2
−7x+6=x
2
⇒x
2
−7x+6=0⇒(x−6)(x−1) =0⇒x=6✓orx=1✗
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 78 / 81

79. Exponential and Logarithmic Functions
Logarithmic & Exponential are Inverses
Example 57
True or False?
“The functionsln (x
2
)and 2lnxhave the same domain !!”
domain ofln
“
x
2
”
=R− {0}domain of 2ln (x)= (0,∞)∴FALSE
Example 58
Find the inverse function off(x) = ln(x+3).
y= ln(x+3)⇒e
y
=e
ln(x+3)
⇒x+3=e
y
⇒x=e
y
−3∴f
−1
(x) =e
x
−3
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 79 / 81

80. Exponential and Logarithmic Functions
Logarithmic & Exponential are Inverses
Example 59
Find the inverse function ofg(x) =e
2x−1
.
y=e
2x−1
⇒lny= ln
“
e
2x−1
”
⇒2x−1= lny⇒x=
1+ lny
2
∴g
−1
(x) =
1+ lnx
2
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 80 / 81

81. Exponential and Logarithmic Functions
One More Example
Example 60
Letf(x) =x
2
−2x−3.
(1) fby shifting the graph ofx
2
.
“
x
2
−2x
”
−3=
“
x
2
−2x+1
”
−1−3= (x−1)
2
−4
(2) f
−1
(x).
y= (x−1)
2
−4x=
p
y+4+1f
−1
(x)=
√
x+4+1
(3) f.
range off= [−4,∞)=domain off
−1
(1,−4)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 81 / 81