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 We may regard as a sum of values of functions f and g given by We may call h t he sum of f and g and denote it by f + g, i.e, h = f + g Thus, Therefore,

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

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

5. Domain of f + g, f – g, f g, and f / g DomainFunction ( Domain of f ) ∩ ( Domain of g ) f + g f – g f g ( Domain of f ) ∩ ( Domain of g ) such that g ( x ) ≠ 0 f / g Example 2. Solution: 0-3 2 3 (d)

6. Class Work 2

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

8. Solution: Example 3: L et f (x ) = x 2 -1 and g ( x ) = 3x + 5. (a) Find ( f ◦ g )( x ) and the domain of f ◦ g. (b) Find ( g ◦ f )( x ) and the domain of g ◦ f. (c) Is f ◦ g = g ◦ f Domain of g = R, Range of g = R, and Domain of f = R Domain of f ◦ g = R In a similar way as in part (a), domain of g◦ f = R

9. Example 4: Let f (x ) = x 2 -1 and g ( x ) = 3x + 5. (a) Find f ( g(2) ) in two different ways: first using the functions f and g separately and second using the composite function f ◦ g (b) Find ( f ◦ f ) ( x ) Solution: First Method g(2) = 3(2) + 5 =11, therefore f ( g(2) ) = f ( 11) = (11) 2 – 1 =121 – 1 = 120 Second Method ( f ◦ g ) ( x ) = 9 x 2 +30 x + 24. Therefore, f ( g (2 ) ) = ( f ◦ g ) ( 2 ) = 9 ( 2) 2 + 30 ( 2 ) + 24 = 120 Same Answer (a) (b) ( f ◦ f ) ( x ) = f ( f ( x ) ) = f( x 2 – 1 ) = ( x 2 – 1 ) 2 - 1= x 4 -2x 2

10. Example 5: ( Finding values of composite functions using tables) Several values of two functions f and g are listed in the following tables. 4321x 1243f ( x ) 4321x 2314g(x) Find ( f◦g)(2) = ( g ◦f ) ( 2 ) = ( f ◦ f ) (2 ) = ( g ◦ g )( 2 ) = Solution: ( f◦g)(2) = f ( g(2) ) = f ( 1) = ( g ◦f ) ( 2 ) = g( f(2 ) ) = g( 4 ) = 3 2 Try to find the rest by yourself ( f ◦ f ) (2 ) = 1 ( g ◦ g )( 2 ) = 4

11. Example 6: ( Finding a composite function form ) Express y = ( 2x + 5 ) 8 in a composite function form Solution: Choice for y = f( u) Choice for u = g(x) Function Value y = ( 2x + 1 ) 8 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 Choice for y = f( x )Choice for u = g(x)Function Value

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 ft 3 / min. And r = r ( t ). Therefore, Substitute V (t) = 4.5 π t

13. The End