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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 V-01 Page 1 e.g.1 Given f (x) = 2x + 1 find: (a) f (3) (b) f (0) (d) f (a) (e) f (2a) (g) f ( t + 1 ) a) 2(3)+1=7 d) 2(a)+1=2a+1 g) 2(t+1)+1=2t+3 (c) f (–1 ) (f ) f (t) b) 2(0)+1=1 e) 2(2a)+1=4a+1 c) 2(-1) +1=-1 f) 2(t)+1=2t+1 End of Block Exercise p.129 Functions V-01 Page 2 . 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: y f ( x ) 2 x Functions V-01 Page 3 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 V-01 Page 4 f (t) e.g.2 The figure shows the graph of the function f (t) given by f (t ) t 2 , 3 t 3 (a) State the domain of the function. [-3, 3] (b) State the range of the function by [0, 9] inspecting the graph. f (t ) t 2 9- -3 -2 -1 0 t 1 2 3 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 End of Block Exercise p.135 Functions V-01 Page 5 Determine the domain of each of the following functions: (a) f x x 5 All real number (b) 1 g t t All real number except 0 2 hs s 2 (c) (d) r x x3 2x 9x 5 2 (, ) (, ) x0 For S≧2 [ 2,) x3 2 x 1x 5 All real number except 5 & -0.5 (, ) x 5 or 0.5 Functions V-01 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 f (t ) 2t 1, g (t ) t 2 f ( g (t )) f (t 2 ) 2t 2 1 2 ( 2 t 1 ) g ( f (t )) g ( 2t 1) Note : f ( g (t )) g ( f (t )) End of Block Exercise p.141 Functions V-01 Page 7 One-to-many rules f x only one y Note: x x is not a function One-to-many rule is not a function. e.g. But functions can be one-to-one or many-to-one. f (x) = 5x +1 is an example of one-to-one function. f (t ) t Functions 2 is an example of many-to-one function. V-01 Page 8 Inverse of a Function x f ( x ) x f f 1 f 1 ( x ) 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 g ( x ) 4 x 3 End of Block Exercise p.148 Functions V-01 Page 9 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 V-01 Page 10 f-1(3x-8)=x --------------------------------- Step 1 Class Exercises Find the inverse functions for the following functions. 1. f(x) = 3x – 8 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 Let Z=8-7x, then x=(8-Z)/7 2. g(x) = 8 – 7x Writing with x instead of Z, then, g-1(x)=(8-x)/7 3. f(x) = (3x – 2)/x Let Z=(3x-2)/x, then x=-2/(Z-3) Writing with x instead of Z, then, f-1(x)=-2/(x-3) Functions V-01 Page 11 TRIGONOMETRIC FUNCTIONS Reference: Croft & Davision, Chapter 9 http://www.math.utep.edu/sosmath L Angles r Two main units of angle measures: degree 90o, 180o radian , 1/2 Unit Conversion radian = 180o e.g.1. Convert 127o in radians. 180 127 x 0.706 rad x 127 180 Trigonometric Functions V-01 r Circle Angle 2 Arc Angle Circumference 2r Arc Length L L r 1 rad Circle Angle 2 360 Page 12 Trigonometric functions y y ; r x cos : r y tan ; x sin P(x,y) r y x x 1 sin 1 sec cos 1 cot tan csc Reference URL: http://home.netvigator.com/~leeleung/sinBox.html Trigonometric Functions V-01 Page 13 The sign of a trigonometric ratio depends on the quadrants in which lies. The sign chart will help you to remember this. y sin’+’ve S T tan’+’ve Trigonometric Functions All ‘+’ve A C x cos ‘+’ve V-01 Page 14 Reference Angle: y II y I O x O 1800 III y y x IV O 180o Trigonometric Functions x O x 3600 V-01 Page 15 Reduction Principle sin sin Where the sign depends on S cos cos T tan tan A C 2 1 =sin (1800-300 ) 30 ○ (b) cos 210 (c) tan 315 2 3 =cos(1800+300) =tan(3600-450) =sin300 =-cos300 =-tan450 =1/2 or 0.5 = =-1 Trigonometric Functions ○ 1 e.g.2 Without using a calculator, find (a) sin 150 45 3/2 V-01 60 1 Page 16 ○ Negative Angles Negative angles are angles generated by clockwise rotations. x sin( ) sin Therefore cos( ) cos tan( ) tan e.g.3 Find (a) sin(-30o) =-sin 30 (b) cos (-300o) o o o =cos(360 - 60 ) o =cos 60 =-1/2 =1/2 Trigonometric Functions V-01 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 V-01 Page 18 Example State the amplitude of each of the following functions: 1. 2. 3. 4. -2≦y ≦2 -4.7≦y ≦4.7 -2/3≦y ≦2/3 -0.8≦y≦0.8 y = 2 sin x y = 4.7cos x y = (2 sin x) / 3 y = 0.8cos x Trigonometric Functions V-01 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 o o sin-1 0.5 = 210 , 330 e.g.5 Solve cos 2 = 0.4 , where 0 2 cos-1(-0.4)= 2 o o 113.58 = 2 or 246.4 = 2 o = 56.8 Trigonometric Functions o or =123.2 V-01 <--- WRONG UNIT 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 S /2, … A 2 =113.580 , 3, … 1.16 rad 1.16 rad 0, 2 … 1.16 rad T Trigonometric Functions V-01 1.16 rad 3 /2, … C 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. sin 2 cos2 1 (1) tan 2 1 sec 2 (2) cot 1 csc (3) 2 Trigonometric Identities 2 V-01 Page 22 Exercise: Derive (2) and (3) from (1) sin cos 1 2 2 sin cos 1 2 cos cos 2 2 2 sin cos 2 sec 2 cos cos 2 tan 2 1 sec 2 2 Trigonometric Identities 2 sin 2 cos 2 1 sin 2 cos 2 sin 2 sin 2 cos 2 2 sin sin 2 1 1 tan 2 cot 2 1 V-01 1 sin 2 csc 2 csc 2 csc 2 Page 23 e.g.1 (a) Solve 2x x 1 0 2 2x -1 x 1 (2x -1) (x +1) = 0 x = ½ or x = -1 2x2 + (x)(-1) + 2x + 1(-1) = 2x2 + x - 1 (b) Using (a), or otherwise, solve 2cos2 1 sin , 0 2 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 V-01 Page 24 e.g.2 Solve 4 sec 2 3 tan 5, 0 2 4(tan 2 1) 3 tan 5 4tanθ 4 tan 2 4 3 tan 5 0 4 tan 3 tan 1 0 2 or -1 4tan2θ – 4tanθ+ tanθ - 1 = 4tan2θ – 3tanθ - 1 (4tanθ+ 1) (tanθ- 1) = 0 tanθ= 1 tanθ 1 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 V-01 Page 25 Solving equations with given identities e.g.3 Using the compound angle formula sin( A B) sin A cos B cos A sin B find the acute angle such that 2 sin 23 cos 2 cos 23 sin 1 2 1 45 ○ 1 2(sin 23 cos cos 23 sin ) 1 30 ○ 2 sin( 23 ) 1 sin( 23 ) 1 2 23 30 3 2 60 1 7 Trigonometric Identities ○ V-01 Page 26 e.g.4 Using the double-angle formula sin 2 A 2 sin A cos A solve sin2 = sin , where 0º <360 º 2 sin cos sin 2 sin cos sin 0 sin (2 cos 1) 0 To make the answer to be 0, either sinθ=0 or 2cosθ-1=0 cosθ=0.5 θ= 0° or 180° θ= 60° or 300° Trigonometric Identities V-01 Page 27 e.g.5 Using the double-angle formula cos 2 A cos 2 A sin 2 A 2 cos 2 A 1 1 2 sin 2 A solve cos2 = sin , where 0 2. 1 2 sin 2 sin 2 sin 2 sin 1 0 (2 sin 1) (sin 1) 0 sinθ= 0.5 or 2sinθ -1 sinθ 1 2sin2θ – sinθ+ 2sinθ - 1 = 2sin2θ + sinθ - 1 sinθ= -1 θ= 30°, 150° or 270° = π/6, 5π/6 or 3 π/2 Trigonometric Identities V-01 Page 28 Engineering waves Reference: Croft &Davison , pp 348 Often voltages and currents vary with time and may be represented in the form A sin( t ) or A cos( t ) 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: i) 2 ii) 5 phase angle=0 (a) y = 2 sin 5t (b) y = sin (t/2) i) Trigonometric Identities 1 ii) ½ V-01 phase angle =0 Page 29 Trigonometric Identities V-01 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 2π/6 =π/3 2. y = 5.6 cosπ t 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 T=π/3, 2. y = 5.6cosπ t T=2, Trigonometric Identities f = 3/π f=½ V-01 Page 31 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 3.8 0.76 5 t 0.93 sin 1 (0.76) 49.46 sin( t 0.93) t 0.93 49.46 180 t 0.93 0.86rad or 2.28rad t 0.07rad or 1.35rad t 6.21rad or 1.35rad What ? –ve rad ??? End of Chapter Exercise: p.360 Trigonometric Identities V-01 Page 32 Formula for Reference (Given in the Exam) sin A B sin A cos B cos A sin B sin A sin B 2 sin A B cos A B 2 2 cos A B cos A cos B sin A sin B tan A tan B tan A B 1 tan A tan B A B A B sin A sin B 2 cos sin 2 2 A B A B cos A cos B 2 cos cos 2 2 A B A B cos A cos B 2 sin sin 2 2 2 sin A cos B sin A B sin A B 2 cos A cos B cos A B cos A B 2 sin A sin B cos A B cos A B Trigonometric Identities V-01 Page 33 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Reference: Croft & Davision, Chapter 8 p.253 http://www.math.utep.edu/sosmath x The exponential function is y e where e = 2.71828182….. y ex y 30 25 Properties (1) e x e y e x y (2) ex e y e x y (3) ( e x ) r e rx 20 (5) e x as x 10 x (6) e 0 as x 5 1 (7) e x 0 for all x (4) e 0 1 Page 34 15 V-01 -3 -2 -1 0 1 2 3 x e.g. 1 Simplify (a) e2 x e x e 3 x b e 4 x (e 2 x 1) 2 e 2 x x ( 3 x ) e 4 x [( e 2 x ) 2 2e 2 x (1) 1] e 6x e (e 2e 1) 4x 4x e 4 x e 4 x 2e 2 x 1 2e 2 x 1 Exercise: p.259 Page 35 2x V-01 Applications : Laws of growth and decay (A)Growth curve y y Ae kx , k 0 A x 0 e.g. Change of electrical resistance (R) with temp. R R0e Page 36 V-01 (B) Decay Curve y A y Ae kx , k 0 x 0 e.g. Discharge of a capacitor q (t ) Qe Exercise: p.259 Page 37 V-01 t CR Class Exercise Q=50, C=0.25 and R=2 a) When t=1, q(t) = ? q(1) = 6.77 b) When R is double, q(1) = ? q(1) = 18.39 Page 38 V-01 Logarithmic Functions If N a x then x loga N , where a 1. The number a is called the base of the logarithm. e.g. 16 2 , 4 log 2 16 4 125 53 , log 5 125 3 0.01 10 2 , log 10 0.01 2 In particular, If N 10 then x log N (common logarithm) x If N e x then x nN (natural logarithm ) Exercise: p.271 Page 39 V-01 Properties of nx (1) nA nB n( AB) (5) nx as x A (2) nA nB n B (6) nx as x 0 (3) nAn nnA (7) domain (0, ) (4) n1 0 (8) range (, ) y nx y 1 1 2 3 4 0 -1 5 x -2 -3 Exercise: p.275 V-01 Page 40 (a) 5 3e 2 x (b) 36 t 72(1 e 3 Solving equations 5 2x e.g.2 e Solve 3 5 ln 2 x 3 1 5 x ln 2 3 x 0.255 Page 41 0.5 1 e 0.5 e 4.92 ) (c) 2.58 n x t 3 2.58 ln 4.92 ln x t 3 ln x ln 4.92 2.58 ln x 0.987 t ln 0.5 3 t 2.079 x 0.373 V-01 e.g.3 The decay of current in an inductive circuit is given by i 50e 0.1t t 0 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. i 50e 0.1t for t (0) i 50e 0.1( 0 ) i 50e 0.1( t ) for i 15 for t 3 i 50e i 50e 0.1t 0.1( 3) i 50 1 i 50 x0.741 i 50 i 37.04 15 50e 0.1t 15 e 0.1t 50 ln 0.3 0.1t t 12.04 Page 42 V-01