FUNCTIONS Reference: Croft & Davision, Chapter 6 p.125 math.utep/sosmath

1. 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'. Page 1

2. 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 Page 2

3. 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: • . Page 3

4. 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 thaty takes is called the range of the function. Page 4

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 Page 5

6. Determine the domain of each of the following functions: All real number All real number except 0 For S≧2 All real number except 5 & -0.5 • (a) • (b) • (c) • (d) Page 6

7. 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 Page 7

8. 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. Page 8

9. 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 Page 9

10. 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 Page 10

11. 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) Page 11

12. L r r 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. Page 12

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

14. y sin’+’ve S All ‘+’ve A x C cos ‘+’ve T tan’+’ve • The sign of a trigonometric ratio depends on the quadrants in which  lies. • The sign chart will help you to remember this. Page 14

15. y II I y     x x O O    III IV y y   x x O  O  • Reference Angle:  Page 15

16. Reduction Principle • Where the sign depends on A S • e.g.2 Without using a calculator, find T C 1 45○ 1 30○ 2 =sin (1800-300 ) =sin300 =1/2 or 0.5 =tan(3600-450) =-tan450 =-1 =cos(1800+300) =-cos300 = 60○ 1 Page 16

17. Negative Angles • 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 Page 17

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

19. 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 Page 19

20. Simple trigonometric equations • Notation : • If sin  = k then  = sin-1k ( sin-1is 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 Page 20

21.  /2, … , 3, … 0, 2 … 3 /2, … cos-1(-0.4) = 2 = 113.580 = 0.631 rad • e.g.5 Solve cos2 =  0.4 , where 0   2 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 A S 2 =113.580 1.16 rad 1.16 rad 1.16 rad 1.16 rad T C Page 21

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) Page 22

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

24. 2x -1 x 1 2x2 + (x)(-1) + 2x + 1(-1) = 2x2 + x - 1 • (2x -1) (x +1) = 0 • x = ½ or x = -1 • e.g.1 (a) Solve • (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 Page 24

25. 4tanθ 1 tanθ -1 4tan2θ – 4tanθ+ tanθ - 1 = 4tan2θ – 3tanθ - 1 • e.g.2 Solve (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 Page 25

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 Page 26

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=0 or 2cosθ-1=0 cosθ=0.5 θ= 60° or 300° θ= 0° or 180° Page 27

28. 2sinθ -1 sinθ 1 2sin2θ – sinθ+ 2sinθ - 1 = 2sin2θ + sinθ - 1 • e.g.5 Using the double-angle formula • solve cos2 = sin  , where 0   2. sinθ= 0.5 or sinθ= -1 θ= 30°, 150° or 270° = π/6, 5π/6 or 3 π/2 Page 28

29. 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 Page 29

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31. The period, T, of both y = A sin ωt 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 Page 31

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 + 0.93 ) • (b) Solve 5 sin ( t + 0.93) = 3.8, where 0  t  2 What ? –ve rad ??? • End of Chapter Exercise: p.360 Page 32

33. Formula for Reference(Given in the Exam) Page 33

34. EXPONENTIAL AND LOGARITHMIC FUNCTIONS 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 0 -2 -1 1 2 3 Page 34

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

36. Applications : Laws of growth and decay (A)Growth curve e.g. Change of electrical resistance (R) with temp.  0 y A x Page 36

37. y • (B) Decay Curve • e.g. Discharge of a capacitor • Exercise: p.259 A x 0 Page 37

38. 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

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

40. Properties of • Exercise: p.275 y 1 1 2 3 4 5 x 0 -1 -2 -3 Page 40

41. Solving equations • e.g.2 Solve Page 41

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. Page 42