Lecture_18 - MA1100 Lecture 18 Functions Composition of...

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1 MA1100 Lecture 18 Functions Composition of functions Inverse functions
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MA1100 Lecture 17 2 Composition of functions R R f Example f: R Ø R defined by f(x) = 3x 2 + 2 g: R Ø R defined by g(x) = sin(x) f(x) = 3x 2 + 2 g( ) g( ) = sin( 3x 2 + 2 ) g(x) = sin(x) f( ) f( ) = 3 sin(x) 2 + 2 R g R R g R f
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MA1100 Lecture 17 3 Composition of functions Definition f: A Ø B and g: B Ø C be functions. The composition of f and g is the function g o f : A Ø C defined by (g o f)(x) = g(f(x)) for all x œ A. We say g o f is a composite function . f(x) x A B f g(f(x)) C g g o f
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MA1100 Lecture 17 4 Composition of functions Remark f: A Ø B and g: B Ø C be functions. f(x) x A B f g(f(x)) C g g o f •f o g may not be defined if A C g o f is not equal to f o g •“ composition of f and g ”notthesameas composition of g and f
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MA1100 Lecture 17 5 Composition of functions Codomain of f domain of g f: A Ø B and g: C Ø D be functions. We can define the composition g o f: A Ø D if range(f) Œ domain of g : g(f(x)) œ D For all x œ A, f(x) œ range(f) f(x) œ domain(g) f(x) has an image in D under g x has an image in D under (g o f)
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MA1100 Lecture 17 6 Composition of functions p q r s t a b c d A B f Example We can define the composite function g o f : A Ø D w x y z D g a b c d A w x y z D g o f p q r s C range(f) Œ domain of g
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MA1100 Lecture 17 7 Composition of functions Example (Cartesian product) f: R 2 Ø R 2 defined by f(x, y) = (x+y, 2y) g: R 2 Ø R 2 defined by g(x, y) = (5y, 3x) What are the composite functions (i) g o f : R 2 Ø R 2 and (ii) f o g : R 2 Ø R 2 ? (g o f)(x, y) = (10y, 3x+3y) (f o g)(x, y) = (3x+5y, 6x)
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MA1100 Lecture 17 8 Composition of functions Example (3 or more functions) f: A Ø B, g: B Ø C, h: C Ø D g o f : A Ø C h o ( g o f ): A Ø D f: A Ø B, g: B Ø C, h: C Ø D h o g : B Ø D ( h o g ) o f : A Ø D f: R Ø R f(x) = x 2 g: R Ø R g(x) = sin x h: R Ø R h(x) = x 1/3 Associative law: h o (g o f) = (h o g) o f h o ( g o f ) = [ sin(x 2 ) ] 1/3 ( h o g ) o f = [sin( x 2 )] 1/3
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Lecture 17 9 Composition of functions Example f: A Ø A f 2 (x) = ( f o f )(x) = f(f(x)) f 2 : A Ø A is the composition of f and itself . f(x) = 3x + 2
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Lecture_18 - MA1100 Lecture 18 Functions Composition of...

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