Reference: Croft & Davision, Chapter 6 p.125 FUNCTIONS Reference: Croft & Davision, Chapter 6 p.125 http://www.math.utep.edu/sosmath 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 Page 1

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 Page 2

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 Page 3

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 Page 4

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 Page 5

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 Page 6

Composition of Functions Reference URL: http://archives.math.utk.edu/visual.calculus/0/compositions.5/ When the output from one function is used as the input to another function - Composite Function Consider End of Block Exercise p.141 Functions Page 7

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 Page 8

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 Page 9

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 Page 10

Find the inverse functions for the following functions. f-1(3x-8)=x --------------------------------- Step 1 Let Z=3x-8, then x=(Z+8)/3 ------------Step 2 And then, f-1(Z) =(Z+8)/3 ----------------Step 3 Writing with x instead of Z, then, f-1(x)=(x+8)/3 ----------------------------Step 4 Class Exercises Find the inverse functions for the following functions. 1. 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 Page 11

Two main units of angle measures: degree 90o, 180o radian  , 1/2  TRIGONOMETRIC FUNCTIONS Reference: Croft & Davision, Chapter 9 http://www.math.utep.edu/sosmath 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 Page 12

Trigonometric functions Reference URL: http://home.netvigator.com/~leeleung/sinBox.html y P(x,y) r  y x x Trigonometric Functions Page 13

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 Page 14

Reference Angle:  II I    IV III y y     x x O O y y   x x O Trigonometric Functions Page 15

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 (1800-300 ) =sin300 =1/2 or 0.5 =cos(1800+300) =-cos300 = =tan(3600-450) =-tan450 =-1 60○ 1 Trigonometric Functions Page 16

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 Page 17

Trigonometric graphs Consider the function y = A sin x, where A is a positive constant. The number A is called the amplitude. Trigonometric Functions Page 18

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 Page 19

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-1 0.5 = 210o, 330o cos-1(-0.4)= 2 113.58o= 2 or 246.4o= 2 = 56.8o or =123.2o <--- WRONG UNIT Trigonometric Functions Page 20

e.g.5 Solve cos2 =  0.4 , where 0    2 cos-1(-0.4) = 2 = 113.580 = 0.631 rad Thus: 2 =  - 1.16;  + 1.16; 3 - 1.16; 3 + 1.16, ….. = 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 =113.580 1.16 rad 1.16 rad 1.16 rad 1.16 rad T C Trigonometric Functions Page 21

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 Page 22

Exercise: Derive (2) and (3) from (1) Trigonometric Identities Page 23

(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= 0.5 or sinθ= -1 θ= 30°, 150°, 270° = π/6, 5π/6, 3π/2 Trigonometric Identities Page 24

(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= 1 or tanθ= - 0.25 θ= 45°, 225°, 166° or 346° = π/4, 5π/4, 0.92π or 1.92π End of Block Exercise: p.336 Trigonometric Identities Page 25

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 Page 26

e.g.4 Using the double-angle formula solve sin2 = sin  , where 0º <360 º To make the answer to be 0, either sinθ=0 or 2cosθ-1=0 cosθ=0.5 θ= 60° or 300° θ= 0° or 180° Trigonometric Identities Page 27

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= 0.5 or sinθ= -1 θ= 30°, 150° or 270° = π/6, 5π/6 or 3 π/2 Trigonometric Identities Page 28

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) 2 ii) 5 phase angle=0 i) 1 ii) ½ phase angle =0 Trigonometric Identities Page 29

Trigonometric Identities Page 30

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 Page 31

∵ 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 + 0.93 ) ∵ max sin θ=1 and min sin θ=-1 ∴ max = 5 and min = -5 (b) Solve 5 sin ( t + 0.93) = 3.8, where 0  t  2 What ? –ve rad ??? End of Chapter Exercise: p.360 Trigonometric Identities Page 32

Formula for Reference (Given in the Exam) Trigonometric Identities Page 33

EXPONENTIAL AND LOGARITHMIC FUNCTIONS http://www.math.utep.edu/sosmath Reference: Croft & Davision, Chapter 8 p.253 http://www.math.utep.edu/sosmath The exponential function is where e = 2.71828182….. Properties y 30 25 20 15 10 5 1 x -3 -2 -1 1 2 3 Page 34

e.g. 1 Simplify Exercise: p.259 Page 35

Applications : Laws of growth and decay (A)Growth curve e.g. Change of electrical resistance (R) with temp.  y A x Page 36

e.g. Discharge of a capacitor y (B) Decay Curve e.g. Discharge of a capacitor Exercise: p.259 y A x Page 37

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 Page 38

Logarithmic Functions The number a is called the base of the logarithm. In particular, Exercise: p.271 Page 39

Properties of Exercise: p.275 y 1 1 2 3 4 5 x -1 -2 -3 Page 40

Solving equations e.g.2 Solve Page 41

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. Page 42