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Objectives 1. Compute operations on functions

Objectives 1. Compute operations on functions 2. Find the composition of two functions and the domain of the composition. Operation on functions. Functions are often defined using sums, differences, products and quotients of various expressions. For example, if.

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Objectives 1. Compute operations on functions

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  1. Objectives 1. Compute operations on functions 2. Find the composition of two functions and the domain of the composition

  2. Operation on functions Functions are often defined using sums, differences, products and quotients of various expressions. For example, if as a sum of values of functions fand g given by We may regard Thus, We may call hthe sumof fand g and denote it by f + g, i.e, h = f + g Therefore,

  3. In general, if f and g are any two functions, we use the terminology and notation given by the following chart Terminology Function Value Sum f + g ( f + g ) ( x ) = f (x ) + g (x ) Difference f – g ( f – g ) ( x) = f ( x ) – g ( x ) Product f g f g ( x ) = f ( x) g ( x ) Quotient Example 1. Solution

  4. Class Work 1 If f( x ) = - x2and g ( x ) = 2x – 1. Find

  5. -3 0 3 2 Domain of f + g, f – g, f g, and f / g Example 2. Solution: (d)

  6. Class Work 2

  7. f(g(x) g( x ) x Composite Functions Definition:The composite function f ◦ g of two functionsfand g is defined by ( f ◦ g )( x ) = f (g(x) ) f ◦ g f g Domain of g Domain of f

  8. Example3: Let f (x ) = x2 -1 and g ( x ) = 3x + 5. (a) Find( f ◦ g )( x ) and the domain off ◦ g. (b) Find( g ◦ f )( x )and the domain of g ◦ f. (c) Isf ◦ g = g ◦ f Solution: Domain of g = R, Range of g = R, and Domain off = R Domain of f ◦ g = R In a similar way as in part (a), domain of g◦ f = R

  9. Example4: Let f (x ) = x2 -1 and g ( x ) = 3x + 5. (a) Findf ( g(2) )in two different ways: first using the functions f and g separately and second using the composite functionf ◦ g (b) Find( f ◦ f ) ( x ) Solution: (a) First Method g(2) = 3(2) + 5 =11, therefore f (g(2) ) = f ( 11) = (11)2 – 1 = 121 – 1 = 120 Same Answer Second Method ( f ◦ g ) ( x ) = 9 x2 +30 x + 24. Therefore, f ( g (2 ) ) = ( f ◦ g ) ( 2 ) = 9 ( 2)2 + 30 ( 2 ) + 24 = 120 (b) f (f ( x ) ) = ( f ◦ f ) ( x ) = f( x2 – 1 ) = ( x2 – 1 )2 - 1= x4 -2x2

  10. Example5: ( Finding values of composite functions using tables) Several values of two functions f and g are listedin the following tables. Find ( f◦g)(2) = ( g ◦f ) ( 2 ) = ( f ◦ f ) (2 ) = ( g ◦ g )( 2 ) = Solution: 3 f(g(2)) = ( f◦g)(2) = f ( 1) = 2 g( f(2) ) = g( 4 ) = ( g ◦f ) ( 2 ) = Try to find the rest by yourself 1 ( f ◦ f ) (2 ) = 4 ( g ◦ g )( 2 ) =

  11. Example 6: ( Finding a composite function form ) Express y = ( 2x + 5 )8 in a composite function form Function Value Choice for u = g(x) Choice for y = f( x ) Solution: Inner function = u Note: y = ( f ◦ g ) ( x ) = f ( g (x) ) = f ( u ) = f ( 2x +1 ) = ( 2x + 1) 8 Class Work Express the following functions in a composite function form

  12. Word Problem using composite Functions Example 7: ( Dimensions of a balloon ) A spherical balloon is being inflated at a rate of 4.5 π ft 3 / min. Express its radius r as a function of time t ( t in minutes ), assuming that r = 0 when t = 0. Solution: At time t , V(t) = 4.5 π t ft3 / min. And r = r ( t ). Therefore, Substitute V(t) = 4.5 π t

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