University of Duhok

College of Engineering
Architectural Department
First Year
𝟏𝒔𝒕 semester

Chapter 1
Functions

Lecturer: Marwa Abdulkareem

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Chapter content:

1.1 Introduction.
1.2 Trigonometric Functions
1.3 Inverse Trigonometric Functions
1.4 the exponential function
1.5 Logarithmic Functions
1.6 hyperbolic functions
1.7 inverse hyperbolic functions

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1.1 Introduction
Cartesian Coordinates in the Plane
P (a , b)
a: is the x-coordinate
b: is the y-coordinate

Example:
Specify the following points on the
x y-axis?

1) Point 1: (0,0)
2) Point 2: (1,0)
3) Point 3: (1,3)
4) Point 4: (-2,1)
5) Point 5: (-2,-1)
6) Point 6: (1,-2)

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 Example:
Determine the slope of L1 and L2
Method of Finding Line Equation: according to their coordinates on
the graph?

A- Slope of Line Solution:

Slope of the nonvertical line P1 P2

∆𝒚 𝒚𝟐 −𝒚𝟏
m= =
∆𝒙 𝒙𝟐 −𝒙𝟏

B- Distance Formula for Points in the Plane

the distance between p1(x1, y1) and p2(x2, y2) is:

d = (∆𝑥)2 +(∆𝑦)2 = (𝑥2 − 𝑥1)2 +(𝑦2 − 𝑦1)2

Example:
find out the distance between p1(1, 6) and P2(5, 9).

Solution:

d = (5 − 1)2 +(9 − 6)2

d = 42 + 32 = 5

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C- Slope – point Equation
The point-slope equation of the line that passes through the
point (x1, y1) and has slope m is :

y = y 1 + m ( x – x1 )

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Mathematical Models:
a- Linear Function b – power functions

Y = mx + b y = 𝒙𝒂
If a = 2, 3;

x Y = 𝒙𝟐 Y = 𝒙𝟑
-2 4 -8
-1 1 -1
0 0 0
1 1 1
2 4 8 9

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c – polynomials functions

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Shifting graphs

Y=𝑥 2 + 1 Y= (𝑥 + 1)2

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Example
Solution:

Position 1: y =(𝒙 − 𝟐)𝟐 +2

Position 2: y = (𝒙 + 𝟐)𝟐 +2
Position 3: y =(𝒙 + 𝟑)𝟐 - 2
Position 4: y = (𝒙 − 𝟏)𝟐 - 4

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 Inverse functions:

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Example 1:
𝟏
Find the inverse of y = x + 1
𝟐

Solution:
𝟏
1. Solve for x in terms of y: y = x + 1
𝟐
𝟏
y= x+1 ∗ (2)
𝟐
2y = x + 2
x = 2y - 2.

2. Interchange x and y: y = 2x - 2.
𝟏
The inverse of the function ƒ(x) = x + 1 is 𝐟 −𝟏 (x) = 2x - 2.
𝟐

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1.2 Trigonometric Functions
Trigonometry is a branch of mathematics that focuses on
relationships between the sides and angles of triangles. The
word trigonometry comes from the Latin derivative of Greek
words for triangle (trigonon) and measure (metron).

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Example:On a circle of radius 10 m, how long is an arc that subtends a
central angle of
• (a) 4π/ 5 radians?
• (b) 110°?

Solution:

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Angles are formed by an initial ray and a terminal ray. An initial

side is said to be in standard position when it’s vertex is located
at the origin and the ray goes along the positive x axis.

If the rotation is in clockwise direction the angle is negative

and it is positive if the rotation is in the anti-clockwise
direction
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The common triangles

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Signs of Trigonometric Functions:

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 Examples: Find the value of
1- sin (𝟑𝟎𝟎° )
1
3
Sin 300° =sin 60° =
2
3
𝟑 2
Sin 𝟑𝟎𝟎° = -
𝟐

2- tan ( 𝟐𝟏𝟎° )

1
tan 𝟐𝟏𝟎° = tan 30° = 3 210°

3 30°
𝟏
tan 𝟐𝟏𝟎° =
𝟑
1
2
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3- csc (−𝟐𝟐𝟓° )
45°
csc (−225° ) = csc (45° ) 2
1 1
csc (45° ) = sin 45
1
csc (45° ) = 1 = 2 −225°
2 1
csc (−𝟐𝟐𝟓° ) = 𝟐

4- sin (−𝟑𝟗𝟎° )
° ° −390°
sin (−390 ) = sin (30 )
1 3
sin (30° ) = 2
30°
1
𝟏 2
sin (−𝟑𝟗𝟎° ) = -
𝟐

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sin 𝑥
x Y = tanx =
cos 𝑥
−𝜋 2 ∞
−𝜋 3 - 1.732
−𝜋 6 - 0.577
0 0
𝜋 6 0.577
𝜋 3 1.732
𝜋 2 ∞
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 1
x Y = csc =
sin 𝑥
−𝜋 ∞
−𝜋 2 −1
−𝜋 4 −1.414
0 ∞
𝜋 4 1.414
𝜋 2 1
𝜋 ∞ 31

General form to find the period and shifts of trigonometric functions.

𝑌 = 𝐴𝑠𝑖𝑛 𝐵𝑥 + 𝑐 + 𝐷
𝟐𝝅
Period: 𝑷= Horizontal shift: 𝐵𝑥 + 𝐶 = 0
𝑩

Upper limit: 𝐷+𝐴 Lower limit: 𝐷−𝐴

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Trigonometric Identities

𝐜𝐨𝐬 𝟐 𝛉 + 𝐬𝐢𝐧𝟐 𝛉 = 𝟏

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1.3 Inverse Trigonometric Functions

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 Example: Graph -

𝒀 = 𝟑𝒔𝒊𝒏−𝟏 𝟐𝒙 3𝜋
2
𝜋
𝒚
= 𝒔𝒊𝒏−𝟏 𝟐𝒙 𝜋 2
𝟑 1
2
−1
−1 ≤ 2𝑥 ≤ 1
−𝜋 𝑦 𝜋 2
≤ ≤ −1 1 −𝜋 2
2 3 2 ≤𝑥≤
2 2 −𝜋
−3𝜋
2

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1.4 the exponential functions

Graphs of exponential functions. 48

 𝒇 𝒙 = 𝒂. 𝒃(𝒙−𝒉) + 𝒌
𝑫: (−∞, ∞ )
𝑹: 𝒌, ∞ 𝒊𝒇 𝒂 > 𝟎
𝑹: −∞, 𝑲 𝒊𝒇 𝒂 < 𝟎

Example : find the domain and range for

𝑥+1
𝑓 𝑥 = −2 +3

𝑫: (−∞, ∞ )
3
𝑹: −∞, 𝟑
2
1

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1.5 Logarithmic Functions :
The function y = ln x is called the natural logarithm function, and
y = log x is often called the common logarithm function.

(1)

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Y = 𝟐𝒙
ln y = ln 2𝑥
ln y = x ln 2 (Interchange x and y)
Ln x = y ln 2
ln 𝑥
Y= ln 2
(apply eq. 1)

Y = 𝒍𝒐𝒈𝟐 𝒙

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 Properties of Logarithms

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Example:

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1.6 hyperbolic functions

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 example Use the definitions and the identity to find the
values of the remaining five hyperbolic functions.

example

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Example: Solve for x the equation

5 cosh x + 3 sinh x = 4.
Solution.
From the basic definitions of sinh and cosh, the equation can be written as

5 cosh x + 3 sinh x = 4.

𝒆𝒙 + 𝒆−𝒙 𝒆𝒙 − 𝒆−𝒙
5 ( ) +3 ( ) =4
𝟐 𝟐
𝟓 𝟑
(𝒆𝒙 + 𝒆−𝒙 ) + (𝒆𝒙 − 𝒆−𝒙 ) = 4
𝟐 𝟐

𝟓 𝟓 𝟑 𝟑
𝒆𝒙 + 𝒆−𝒙 + 𝒆𝒙 - 𝒆−𝒙 = 4
𝟐 𝟐 𝟐 𝟐

𝟖 𝟐
𝒆𝒙 + 𝒆−𝒙 = 4
𝟐 𝟐

4 𝒆𝒙 + 𝒆−𝒙 = 4 ( multiply by 𝒆𝒙 ) 60
 (4 𝒆𝒙 ) 𝒆𝒙 + (𝒆−𝒙 )𝒆𝒙 = 4 𝒆𝒙

4 𝒆𝒙+𝒙 + 𝒆−𝒙+𝒙 = 4𝒆𝒙 4𝒆𝟐𝒙 + 𝒆𝟎 = 4 𝒆𝒙

4𝒆𝟐𝒙 + 1 = 4𝒆𝒙
4 (𝒆𝒙 )𝟐 - 4 𝒆𝒙 + 1 = 0
a = 4, b = -4, c = 1

−𝒃 ± 𝒃𝟐 −𝟒𝒂𝒄 𝟒± 𝟏𝟔−𝟒∗𝟒∗𝟏 𝟒
𝒆𝒙 = = =
𝟐𝒂 𝟐 ∗𝟒 𝟖

𝟏
𝒆𝒙 = ( add ln to both sides of equation)
𝟐

𝟏
ln 𝒆𝒙 = ln
𝟐

𝟏
X = ln = - 0.693
𝟐

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Graphs of Hyperbolic Equations:

𝑒 𝑥 −𝑒 −𝑥 x 𝑠𝑖𝑛ℎ𝑥 1
x Y=sinh x =
2
Y= tanh x = 𝑐𝑜𝑠ℎ𝑥 Y= coth x = 𝑡𝑎𝑛ℎ𝑥
-1 -1.175 -1 - 0.761 -1.313
0 0 0 0 ∞
1 1.175 1 0.761 1.313

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 Y=coshx Y=sinhx

1 1
𝑦 = 𝑠𝑒𝑐ℎ𝑥 = 𝑦 = 𝑐𝑠𝑐ℎ𝑥 =
𝑐𝑜𝑠ℎ𝑥 𝑠𝑖𝑛ℎ𝑥

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Inverse of Hyperbolic Functions:

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Inverse Hyperbolic Functions :

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Prove that:

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Prove that:

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