Section 2.6 Combinations of Functions: Composite FunctionsPowerPoint PresentationSlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Section 2.6Combinations of Functions: Composite FunctionsThe Domain of a Function( ) ( )( ) ( )( ) ( ) ( )223x+5g(x)=x 4 5x 4 5 05 1 05 0 1 0 x=5 x=-1: - ,-1 1,5 5,xxx xx xDomain- -- - =- + =- = + =� �-ȥ)( ) 2 52 5 0 2x 55 x25: ,2h x xxDomain= -- �������( )2( ) 7 - ,f x x xDomain= -��Domains of Other Functions( )( ) 4 5: ,f x xDomain= -- ��( )3( ): ,f x xDomain=- ��ExampleFind the domain of the function4x-1f(x)=3x+2ExampleFind the domain of the functionf(x)= 4x-1The Algebra of Functions( ) ( )( )( ) ( )( )22fIf the function can begsimplified, determine the domainbefore simplifying.Example;f(x)= x 4 and g(x)=x-2 f fx 2 in ; Domain of : ,2 2,g g2 2f 42g 2 2x xxx xx x-�-�ȥ- +� �-= = = +� �- -� �The domain of f+g is the set of all real numbers that are common to the domain of f and the domain of g. Thus we must find the domains of f and g beforefinding their intersection.Suppose ( ) x+3 af x = nd g(x)= x-2 then(f+g)(x)= x+3 2Now for their domains.( ) x+3 g(x)= x-23 0 x-2 0 3 x 2So the domain for the sum of the functionsis x 2 which in interxf xxx+ -=+ � ��- ��[)val notation is 2,�Continued on next slideDetermining Domains When Adding or Subtracting FunctionsContinuation of the same problem. xy[)The graph of (f+g)(x)= x+3 2confirms that the domain of this function is 2,x+ -�The domain of f g is the set of all real numbers that are common to the domain of f and the domain of g. Thus we must find the domains of f and g beforefinding their intersection.5Suppose ( ) anxf x�=( )3d g(x)= thenx-215(fg)(x)= x x-2Now for their domains.5 3( ) g(x)=x x-20 x-2 0 x 2So the domain for the product of the functionsis x 0, x 2 which in f xx=� ��� �( ) ( ) ( )interval notation is - ,0 0,2 2,��ȥ xyDetermining Domains when Multiplying FunctionsExample2If f(x)=5x-1 g(x)=x 2 1 Find each of the following:(f+g)(x)(f-g)(x)(fg)(x)f( )gxx- +� �� �� �Example2If f(x)=5x-1 g(x)=5x 9 2 Find the domain of the following:(fg)(x)f( )gxx+ -� �� �� �Example1 1If f(x)= g(x)= Find the domain 2 1of the following:(fg)(x)f( )gx xx-� �� �� �ExampleIf f(x)= x-1 g(x)= x-6 Find the domain for:(f-g)(x)Composite Functionsf(g(x))=0.85x - 300We read this equation as "f of g of x is equal to 0.85x-300." We call f(g(x)) the composition of the function f with g, or a composite function. This composite function is written f go( ) ( ) ( )The domain of f g is ,0 0,3 3,-��ȥoExample( ) ( )3 2Given f(x)= and g(x)= . x-4 xa. Find f g b. Find the domain of f gxo oExample( ) ( )2Given f(x)= and g(x)= x. x-3a. Find f g b. Find the domain of f gxo oDecomposing FunctionsExample( )42Express h(x) as a composition of two functions:( ) 6 5 h x x x= - +Example2Express h(x) as a composition of two functions:1( ) 9 64h xx=-(a) (b)(c)(d)2Find the domain of the function3x-1f(x)=x 6 7x- -( ) ( ) ( )( ) ( ) ( )(] [)(] [ ] [), 1 1,7 7,,1 1,7 7,, 1 7,, 1 1,7 7,-�-�-ȥ-��ȥ-�-ȥ-�-�-ȥ(a) (b)(c)(d)2 If f(x)=3x-1 and g(x)=x ,Find (f+g)(x)22223 1(3 1)3 13 1xxx xx x--+ -- - +(a) (b)(c)(d)( ) ( )Find the domain of f gif f(x)= x-4 and g(x)=4 3xx-o[)[)(](]0,3,, 3,0�- �- �--
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