1. Reference: Croft & Davision, Chapter 6 p.125
FUNCTIONS
Reference: Croft & Davision, Chapter 6 p.125
Basic Concepts of Functions
A function is a rule which operates on an input and produces a single output from that input.
Consider the function given by the rule: 'double the input'.
Functions
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2. a) 2(3)+1=7 b) 2(0)+1=1 c) 2(-1) +1=-1
e.g.1 Given f (x) = 2x + 1 find: (a) f (3) (b) f (0) (c) f (–1 ) (d) f (a) (e) f (2a) (f ) f (t) (g) f ( t + 1 )
a) 2(3)+1=7 b) 2(0)+1=1 c) 2(-1) +1=-1
d) 2(a)+1=2a+1 e) 2(2a)+1=4a+1 f) 2(t)+1=2t+1
g) 2(t+1)+1=2t+3
End of Block Exercise
p.129
Functions
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3. A function may be represented in graphical form.
The Graph of a Function
A function may be represented in graphical form.
The function f (x) = 2x is shown in the figure.
We can write:
Functions
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4. The set of values that y takes is called the range of the function.
In the function y = f (x), x is the independent variable and y is the dependent variable.
The set of x values used as input to the function is called the domain of the function
The set of values that y takes is called the range of the function.
Functions
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5. e.g.2 The figure shows the graph of the function f (t) given by
(a) State the domain of the function.
(b) State the range of the function by inspecting the graph.
End of Block Exercise
p.135
[-3, 3]
[0, 9]
e.g.3 Explain why the value t = 0 must be excluded from the domain of the function f (t) = 1/t.
∵ 1/0 is undefined
Functions
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6. Determine the domain of each of the following functions:
(b)
(c)
(d)
All real number
All real number except 0
For S≧2
All real number except 5 & -0.5
Functions
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7. Composition of Functions Reference URL:
When the output from one function is used as the input to another function - Composite Function
Consider
End of Block Exercise
p.141
Functions
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8. One-to-many rule is not a function.
One-to-many rules
Note:
e.g.
One-to-many rule is not a function.
But functions can be one-to-one or many-to-one.
f (x) = 5x +1 is an example of one-to-one function.
is an example of many-to-one function.
Functions
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9. e.g.4 Find the inverse function of
Inverse of a Function
is the notation used to denote the inverse function of f (x). The inverse function, if exists, reverse the process in f (x).
e.g.4 Find the inverse function of
End of Block Exercise
p.148
Functions
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10. Let Z = 4x-3, and transpose this to give x = (Z+3)/4 Then,
Solution of e.g.4 :
The inverse function, g-1, must take an input 4x – 3 and give an output x. That is,
g-1(4x-3) = x
Let Z = 4x-3, and transpose this to give x = (Z+3)/4
Then,
g-1(Z) = (Z+3)/4
Writing with x as its argument instead of Z gives
g-1(x) = (x+3)/4
Functions
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11. Find the inverse functions for the following functions.
f-1(3x-8)=x Step 1
Let Z=3x-8, then x=(Z+8)/ Step 2
And then, f-1(Z) =(Z+8)/ Step 3
Writing with x instead of Z, then,
f-1(x)=(x+8)/ Step 4
Class Exercises
Find the inverse functions for the following functions.
f(x) = 3x – 8
2. g(x) = 8 – 7x
3. f(x) = (3x – 2)/x
Let Z=8-7x, then x=(8-Z)/7
Writing with x instead of Z, then,
g-1(x)=(8-x)/7
Let Z=(3x-2)/x, then x=-2/(Z-3)
Writing with x instead of Z, then,
f-1(x)=-2/(x-3)
Functions
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12. Two main units of angle measures: degree 90o, 180o radian  , 1/2 
TRIGONOMETRIC FUNCTIONS
Reference: Croft & Davision, Chapter 9
Angles
Two main units of angle measures:
degree 90o, 180o
radian  , 1/2 
Unit Conversion  radian = 180o
e.g.1. Convert 127o in radians. r L
Trigonometric Functions
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13. Trigonometric functions
Reference URL: y
P(x,y) r  y x x
Trigonometric Functions
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14. The sign chart will help you to remember this.
The sign of a trigonometric ratio depends on the quadrants in which  lies.
The sign chart will help you to remember this. y x
All ‘+’ve A
sin’+’ve S T
tan’+’ve C
cos ‘+’ve
Trigonometric Functions
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15. Reference Angle:  II I    IV III y y     x x O O y y   x x O
Trigonometric Functions
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16. S A T C =sin (1800-300 ) =sin300 =1/2 or 0.5 =cos(1800+300) =-cos300 =
Reduction Principle
Where the sign depends on S A T C 1
45○ 1
e.g.2 Without using a calculator, find
30○ 2
=sin ( )
=sin300
=1/2 or 0.5
=cos( )
=-cos300 =
=tan( )
=-tan450
=-1
60○ 1
Trigonometric Functions
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17. Negative angles are angles generated by clockwise rotations.
Therefore
e.g.3 Find (a) sin(-30o) (b) cos (-300o) x 
=cos(360o- 60o)
=cos 60o
=1/2
=-sin 30o
=-1/2
Trigonometric Functions
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18. Trigonometric graphs
Consider the function y = A sin x, where A is a positive constant. The number A is called the amplitude.
Trigonometric Functions
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19. State the amplitude of each of the following functions: 1. y = 2 sin x
Example
State the amplitude of each of the following functions:
1. y = 2 sin x
2. y = 4.7cos x
3. y = (2 sin x) / 3
4. y = 0.8cos x
-2≦y ≦2
-4.7≦y ≦4.7
-2/3≦y ≦2/3
-0.8≦y≦0.8
Trigonometric Functions
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20. Simple trigonometric equations Notation :
If sin  = k then  = sin-1k ( sin-1 is written as inv sin or arcsin). Similar scheme is applied to cos and tan.
e.g.4 Without using a calculator, solve sin  =  0.5,
where 0o    360o
e.g.5 Solve cos 2 =  0.4 , where 0    2
sin = 210o, 330o
cos-1(-0.4)= 2
113.58o= 2 or o= 2
= 56.8o or =123.2o
<--- WRONG UNIT
Trigonometric Functions
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21. e.g.5 Solve cos2 =  0.4 , where 0    2
cos-1(-0.4) = 2 = = 0.631 rad
Thus: 2 =  ;  ; 3 ; 3 , …..
= 1.98, 4.3, 8.26, 10.58, ……
Thus:  = 0.99, 2.15, 4.13 or 5.29 rad
3 /2, …
, 3, …
 /2, …
0, 2 … S A
2 =
1.16 rad
1.16 rad
1.16 rad
1.16 rad T C
Trigonometric Functions
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22. TRIGONOMTRIC EQUATIONS
Reference: Croft & Davision, Chapter 9, Blocks 5, 6, 7
Some Common Trigonometric Identities
A trigonometric identity is an equality which contains one or more trigonometric functions and is valid for all values of the angles involved.
e.g (1)
(2)
(3)
Trigonometric Identities
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23. Exercise: Derive (2) and (3) from (1)
Trigonometric Identities
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24. (b) Using (a), or otherwise, solve
e.g.1 (a) Solve 2x -1 x 1
2x2 + (x)(-1) + 2x + 1(-1)
= 2x2 + x - 1
(2x -1) (x +1) = 0
x = ½ or x = -1
(b) Using (a), or otherwise, solve
2(1-sin2θ) - sinθ-1 = 0
2 - 2sin2θ- sinθ-1 = 0
2sin2θ+ sinθ-1 = 0
sinθ= or sinθ= -1
θ= 30°, 150°, 270°
= π/6, 5π/6, 3π/2
Trigonometric Identities
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25. (4tanθ+ 1) (tanθ- 1) = 0 tanθ= 1 or tanθ= - 0.25
e.g.2 Solve
4tanθ 1
tanθ -1
4tan2θ – 4tanθ+ tanθ - 1
= 4tan2θ – 3tanθ - 1
(4tanθ+ 1) (tanθ- 1) = 0
tanθ= or tanθ=
θ= 45°, 225°, 166° or 346°
= π/4, 5π/4, 0.92π or 1.92π
End of Block Exercise: p.336
Trigonometric Identities
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26. Solving equations with given identities
e.g.3 Using the compound angle formula
find the acute angle  such that 1
45○ 1
30○ 2
60○ 1
Trigonometric Identities
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27. e.g.4 Using the double-angle formula
solve sin2 = sin  , where 0º <360 º
To make the answer to be 0,
either sinθ= or cosθ-1=0
cosθ=0.5
θ= 60° or 300°
θ= 0° or 180°
Trigonometric Identities
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28. e.g.5 Using the double-angle formula
solve cos2 = sin  , where 0   2.
2sinθ -1
sinθ 1
2sin2θ – sinθ+ 2sinθ - 1
= 2sin2θ + sinθ - 1
sinθ= or sinθ= -1
θ= 30°, 150° or 270°
= π/6, 5π/6 or 3 π/2
Trigonometric Identities
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29. 2 ii) 5 phase angle=0 i) 1 ii) ½ phase angle =0 Engineering waves
Reference: Croft &Davison , pp 348
Often voltages and currents vary with time and may be represented in the form
where A : Amplitude of the combined wave
 : Angular frequency (rad/sec) of the combined wave (Affect wave width)
 : Phase angle (left and right movement)
t : time in second
Example
State (i) the amplitude and (ii) the angular frequency of the following waves:
(a) y = 2 sin 5t
(b) y = sin (t/2)
ii) 5 phase angle=0
i) ii) ½ phase angle =0
Trigonometric Identities
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30. Trigonometric Identities
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31. The period, T, of both y = A sin ω t and y = A cos ω t is given by T = (2π)/ω
Example
State the period of each of the following functions:
1. y = 3 sin 6t
2. y = 5.6 cosπ t
2π/6 =π/3
2π/π = 2
The frequency, f, of a wave is the number of cycles completed in 1 second. It is measured in hertz (Hz).
T = 1 / f
Example
State the period and frequency of the following waves:
1. y = 3 sin 6 t
2. y = 5.6cosπ t
T=π/3, f = 3/π
T=2, f = ½
Trigonometric Identities
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32. ∵ max sin θ=1 and min sin θ=-1 ∴ max = 5 and min = -5
E.g. 6 (a) Find the maximum and minimum value of
5 sin ( t )
∵ max sin θ=1 and min sin θ=-1
∴ max = and min = -5
(b) Solve 5 sin ( t ) = 3.8, where 0  t  2
What ? –ve rad ???
End of Chapter Exercise: p.360
Trigonometric Identities
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33. Formula for Reference (Given in the Exam)
Trigonometric Identities
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34. EXPONENTIAL AND LOGARITHMIC FUNCTIONS http://www.math.utep.edu/sosmath
Reference: Croft & Davision, Chapter 8 p.253
The exponential function is
where e = …..
Properties y 30 25 20 15 10 5 1 x -3 -2 -1 1 2 3
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35. e.g. 1 Simplify
Exercise: p.259
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36. Applications : Laws of growth and decay (A)Growth curve
e.g. Change of electrical resistance (R) with temp.  y A x
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37. e.g. Discharge of a capacitor y
(B) Decay Curve
e.g. Discharge of a capacitor
Exercise: p.259 y A x
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38. Q=50, C=0.25 and R=2 a) When t=1, q(t) = ? q(1) = 6.77
Class Exercise
Q=50, C=0.25 and R=2
a) When t=1, q(t) = ?
b) When R is double, q(1) = ?
q(1) = 6.77
q(1) = 18.39
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39. Logarithmic Functions
The number a is called the base of the logarithm.
In particular,
Exercise: p.271
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40. Properties of
Exercise: p.275 y 1 1 2 3 4 5 x -1 -2 -3
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41. Solving equations
e.g.2 Solve
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42. e.g.3 The decay of current in an inductive circuit is given by
Find (a) the current when t=0;
(b) the value of the current when t=3;
(c) the time when the value of the current is 15.
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