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# Chapter 1 (functions).

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### Chapter 1 (functions).

1. 1. BMM 104: ENGINEERING MATHEMATICS I Page 1 of 13 CHAPTER 1: FUNCTIONSRelations  Types of relations :  One – to – one  one – to – many  many – to – oneOrdered Pairs f(x ) .a .c • ( a , c ) and ( b , d ) are known as ordered pairs . • The set of ordered pairs is { ( a , c ) , ( b , d ) } . .b .d • c and d are called the image of the corresponding first component .domain codomain
2. 2. BMM 104: ENGINEERING MATHEMATICS I Page 2 of 13Functions  Definition Function A relation in which every element in the domain has a unique image in the codomain.  Notation of functions : A function f from x to y : f : x  y or y = f ( x ) f(x ) .a .c .b .ddomain codomain  Domain – set of input values for a function Range – the corresponding output values – is a subset of codomain  Elements of domain { a , b } Elements of codomain { c , d }NOTE: Vertical line test can be used to determine whether a relation is a function ornot. A function f ( x ) can have only one value f ( x ) for each x in its domain, so novertical line can intersect the graph of a function more than once.Example: Determine which of the following equations defines a function y in terms of x.Sketch its graph.(i) y + 2 x = 1 (ii) y = 3 x 2 (iii) x 2 + y 2 = 1Domain and RangeThe set D of all possible input values is called the domain of the function.The set of all values of f ( x ) as x varies throughout D is called the range of thefunction.Example:
3. 3. BMM 104: ENGINEERING MATHEMATICS I Page 3 of 13Combining FunctionsSum, Differences, Products, and QuotientsExample: Attend lecture.Composite FunctionsIf f and g are functions, the composite function f  g (“f composed with g”) is defined by ( f  g )( x ) = f ( g ( x )).The domain of f  g consists of the numbers x in the domain of g for which g(x) lies inthe domain of f.Example: Attend lecture.
4. 4. BMM 104: ENGINEERING MATHEMATICS I Page 4 of 13Inverse FunctionsOne-to-One FunctionsA function f(x) is one-to-one if every two distinct values for x in the domain, x1 ≠ x 2 ,correspond to two distinct values of the function, f ( x1 ) ≠ f ( x 2 ) .Properties of a one-to-one function (f −1  f )( x ) = x and ( f  f )( y ) = y −1NOTE: A function y = f ( x ) is one-to-one if and only if its graph intersects eachhorizontal line at most once.Inverse FunctionsFinding the Inverse of a FunctionStep 1: Verify that f(x) is a one-to-one function.Step 2: Let y = f(x).Step 3: Interchange x and y.Step 4: Solve for y.Step 5: Let y = f −1 ( x ) .Step 6: Note any domain restrictions on f −1 ( x ) .NOTE: −1  Domain of f = Range of f −1  Range of f = Domain of fExample:1. Find the inverse of the function f ( x ) = 2 x − 3 .2. Find the inverse of the function f ( x ) = x 3 + 2 .3. Find the inverse of the function f ( x ) = x + 2 . 24. Find the inverse of the function f ( x ) = , x ≠ −3 . x +3Even Function, Odd Function
5. 5. BMM 104: ENGINEERING MATHEMATICS I Page 5 of 13A function y = f ( x ) is an even function of x if f ( −x ) = f ( x ) , odd function of x if f ( −x ) = − f ( x ) ,for every x in the function’s domain. Even Function (Symmetric about the y-axis) Odd Function (Symmetric about the origin)Example: Attend lecture.Exponential functions
6. 6. BMM 104: ENGINEERING MATHEMATICS I Page 6 of 13Function of the form f ( x) = a x , a ≠ 1where a is positive constant is the general exponential function with base a and x asexponent.The most commonly used exponential function, commonly called natural exponentialfunction is y =ex or y = exp( x )where the base e is the exponential constant whose value is e ≈ 2.718281828...Rules for exponential functions i. e x .e y = e x + y ~ Product Rule ex ii. y = e x−y ~ Quotient Rule e iii. (e ) x y ( ) = e xy = e y x ~ Power Rule 1 1 iv. x = e − x or −x = ex ~ Reciprocal Rule e e v. e0 = 1Example:Solve the following exponent equations.(a) 2x 2 +3 = 16 (b) 2 x 3 x +1 = 108 (c) ( 2) x2 = 8x 4Logarithmic Functions
7. 7. BMM 104: ENGINEERING MATHEMATICS I Page 7 of 13The logarithm function with base a, y = log a x , is the inverse of the base a exponentialfunction y = a x ( a > 0 , a ≠ 1) .The function y = ln x is called the natural logarithm function, and y = log x is oftencalled the common logarithm function. For natural logarithm, y = ln x ⇔ ey = xAlgebraic properties of the natural logarithmFor any numbers b > 0 and x > 0 , the natural logarithm satisfies the following rules:1. Product Rule: ln bx = ln b + ln x b2. Quotient Rule: ln = ln b − ln x x3. Power Rule: ln x r = r ln x 14. Reciprocal Rule: ln = −ln x xInverse Properties for a x and log a x1. Base a: a log a x = x , log a a x = x , a > 0 , a ≠ 1, x > 02. Base e: e ln x = x , ln e = x , x x >0Change Base FormulaEvery logarithmic function is a constant multiple of the natural logarithm. ln x log a x = ( a > 0 , a ≠ 1) ln aNOTE: log a 1 = 0 .Example 1: Rewrite the following expression in terms of logarithm.i. 32 = 2 5 ii. 1000 = 10 3 1iii. 0.001 = 10 −3 iv. 3 = 92Example 2: i. Evaluate log 2 3 .
8. 8. BMM 104: ENGINEERING MATHEMATICS I Page 8 of 13 1 2 ii. Simplify log 2 8 − log 2 . 3 7  1 iii. Simplify log 2   + log 2 128 8  10 x  iv. Expand ln 2   y   Example 3: Solve the following equations: 1. 39 = e x 2. 10 x = 0.32 3. ln 2 x = 1.36 4. log ( 3 x − 6 ) = 0.76 5. log 3 ( 2 x + 1) − 2 log 3 ( x − 3 ) = 2 6. 6 3 x +2 = 200Trigonometric FunctionsExample: Attend lecture.Hyperbolic FunctionsHyperbolic functions are formed by taking combinations of the two exponential functionse x and e −x .The six basic hyperbolic functions e x − e −x1. Hyperbolic sine of x: sinh x = 2 e x + e −x2. Hyperbolic cosine of x: cosh x = 2 ex − e −x3. Hyperbolic tangent of x: tanh x = ex + e −x ex + e −x4. Hyperbolic cotangent of x: coth x = x e − e −x 25. Hyperbolic secant of x: sec hx = x e + e −x 26. Hyperbolic cosecant of x: cos echx = x e − e −xShifting a Graph of a Function
9. 9. BMM 104: ENGINEERING MATHEMATICS I Page 9 of 13
10. 10. BMM 104: ENGINEERING MATHEMATICS I Page 10 of 13
11. 11. BMM 104: ENGINEERING MATHEMATICS I Page 11 of 13Example: Attend lecture. PROBLET SET: CHAPTER 11. Let f ( x ) = x 2 − 3 x and find each of the following: 1  1 (a) f ( −3 ) (b) f ( 5 ) (c) f ( 0 ) (d) f   (e) f  −  2  2 (f) f ( a ) (g) f ( 2 x ) (h) 2 f ( x ) (i) f ( x + 3) (j) f ( x ) + f ( 3 ) 1 1 (k) f   (l) x f(x) 12. Let g ( x ) = . Evaluate and simplify the difference quotient: x g( 4 + h ) − g( 4 ) ,h ≠ 0 h3. Find the domain and range of the function defined by each equation. y = ( x − 1) + 2 3 (a) (e) y = 5x 2 3 (b) y = 7 −x (f) y= x +1 x (c) y = −x 2 + 4 x − 1 (g) y= x +3 x −2 (d) y = 3x + 5 (h) y= x +34. Find f  g and g  f . (a) f ( x ) = x 2 ; g( x ) = x − 1 (e) f ( x ) = x + 1 ; g( x ) = x 4 − 1 x +1 (b) f ( x ) = x −3 ; g ( x ) = 2 x +3 (f) f ( x ) = 2 x 3 − 1; g ( x ) = 3 2 x x +3 (c) f(x)= ; g( x ) = (g) f(x)= x ; g( x ) = 4 x −2 x 1 (d) f ( x ) = x 3 ; g( x ) = (h) f ( x ) = 3 1 − x ; g ( x ) = 1 − x 3 x +1 3
12. 12. BMM 104: ENGINEERING MATHEMATICS I Page 12 of 135. Find the inverse g of the given function f, and state the domain and range of g. y = ( x + 1) ; x ≥ −1 2 (a). (e). y = x 2 − 4; x ≥ 0 (b). y = x 2 − 4 x + 4; x ≥ 2 (f). y = 4 − x 2 ;0 ≤ x ≤ 2 1 (c). y= (i). y = x 2 − 4 x; x ≥ 2 x (d). y =− x (j). y = 4x − x2 ; x ≥ 26. Determine whether the following functions are odd, even or neither even nor odd. 3 (a). f ( x) = − 2x (f). f ( x ) =−−8t + −7t x2 (b). (1 − x ) 3 (g). f ( x) = 3 f ( x) = 3 x 3 − 3 x x cos x − x sin x − ( cos x ) 2 (c). f ( x) = 2 (h). f ( x) = 3−x cot x sec x x2 (d). f ( x ) = 3 x 4 sin x (i). f ( x) = cos x + +5 1− x4 x + x2 3 − 1  2 2 (e). f ( x ) =x 3  −x 3 1   (j). f ( x) = tan x +   sin x ANSWERS FOR PROBLEM SET: CHAPTER 1 5 71. (a) 18 (b) 10 (c) 0 (d) − (e) 4 4 (f) a 2 − 3a (g) 4 x 2 − 6 x (h) 2 x 2 − 6 x (i) x 2 + 3 x (j) x 2 − 3 x 1 3 1 (k) 2 − (l) 2 x x x − 3x 12. − 4( 4 + h )3. (a) D = ℜ; R = ℜ (e) D = ℜ R = [0 , ∞) ; (b) D = ( − ∞7 ]; R = [0 , ∞ , ) (f)D = ( − ∞,−1) ∪ ( 1, ∞) ; R = ( − ∞,0 ) ∪ ( 0 , ∞) (c) D = ℜ R = ( −∞,3] ; (g)D = ( − ∞,−3 ) ∪ ( − 3 , ∞) ; R = ( − ∞,1) ∪ ( 1, ∞) (d) D = ℜ; R = ℜ (h)D = ( − ∞,−3 ) ∪ ( − 3 , ∞) ; R = ( − ∞,1) ∪ ( 1, ∞) f  g = ( x − 1) ; g  f = x 2 − 1 24. (a) (b) f  g = 2 x ; g  f = 2 x −3 +3
13. 13. BMM 104: ENGINEERING MATHEMATICS I Page 13 of 13 x +3 6 (c) f g = ;g  f = 4 − 3−x x 1 1 (d) f g = − 1; g  f = ( x +1 3 3 ) 3 x −1 +1 3 ( ) (e) f  g = x ; g  f = x + 2x 2 2 (f) f  g = x; g  f = x (g) f  g = 2; g  f = 4 (h) f  g = x; g  f = x5. (a). g( x ) = x −1 D : x ≥ 0 ; R = y ≥ −1 (b). g( x ) = x + 2 D : x ≥ 0; R = y ≥ 2 1 (c). g( x ) = D : x > 0; R = y > 0 x2 (d). g( x ) = x 2 D : x ≤ 0; R = y ≥ 0 (e). g( x ) = x + 4 D : x ≥ −4 ; R = y ≥ 0 (f). g( x ) = 4 − x D : 0 ≤ x ≤ 4; R = 0 ≤ y ≤ 26. (a). Neither even nor odd (f) Even (b) Neither even nor odd (g) Odd (c) Neither even nor odd (h) Neither even nor odd (d) Odd (i) Even (e) Odd (j) Odd