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Al 2. Section 7.7 Operations on Functions Objective: •

Find the sum, difference, product, and quotient of Functions

•

Find the composition of functions

1) If f and g are functions and x is in the domain of each function, then Sum

Difference

( f + g)(x) = f (x) + g(x) Product

( fg)(x) = f (x) ! g(x)

( f ! g)(x) = f (x) ! g(x)

Quotient ( f / g)(x) = f (x) / g(x), provided g(x) ! 0

Example 1: Given that f ( x ) = x 2 ! 3x + 1 and g ( x ) = 2x + 3 , find each of the following. a) (f + g)(x) b) (f - g)(x) c) ( fg)(x) = d) ( f / g)(x) =

Example 2:Given that f(x) = x2-1 and g(x) = 2x + 5, find each of the following. a) (f + g)(x) b) (f ! g)(x) c) (f/g)(x) 2) The Composition of Functions The composition of functions occurs when a function is performed, and then a second function is performed on the result of the first.

The composite function f !g, is defined as ( f !g )( x ) = f ( g ( x )) , where x is in the domain of g and g(x) is in the domain of f. Range of g is a subset of the domain of f.


! Example 3: The composition of functions can be shown by mappings. Suppose f = {( 3, 4 ) , ( 2, 3) , ( !5,0 )} and g = {( 3,!5 ) , ( 4, 3) , ( 0,2 )} Find:

and

fog

gof

Example 4: Given f ( x ) = x + 3 and g ( x ) = x 2 + x ! 1 a. Find [ fog ]( x ) and [ gof ]( x ) .

b. Evaluate [ fog ]( 2 ) and [ gof ]( 2 ) .

Example 5: If f ( x ) = 2x and g ( x ) = 3x + 1 , find: a. f !" g (1) #$

b. g "# f ( !2 ) $%

Section 7.7 HW: page 387, #s 17-45 odd, 48; opt. 18, 24, 32, 38, 44


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