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Week 4 Composition Function Presentation

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Week 4 3.4 and 3.7 3.4 HW 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 93, 97 For two functions f(x) and g(x) with real number outputs, the following equations are true: ( f + g ) (x) = f(x) + g(x) ( f - g ) (x) = f(x) - g(x) ( fg ) (x) = f(x) g(x) ( f / g ) (x) = f(x) / g(x), where g(x) does not equal 0 Example 1… Find and simplify the functions (g-f)(x) and (g/f)(x), given f(x) = x - 1 and g(x) = x2 - 1. Are they the same function? (g - f)(x) = g(x) - f(x) = x2 - 1 - ( x - 1) = x2 - 1 - x + 1 = x2 - x = x ( x - 1) (g/f)(x) = g(x) / f(x) = ( x2 - 1 ) / ( x - 1 ) = ( (x-1)(x+1) ) / ( x - 1 ) = x + 1 No, these are not the same function Composition Function Combining functions so that the output of one function becomes the input of another function; binary operation that combines the functions (f∘g)(x)=f(g(x)) This can be read at “ f of g of x “ Always solve the inside function first and then input the output of that function into the outside function NOTE: f(g(x)) is almost never equal to g(f(x)) Example 2 Using the functions provided, find f ( g(x) ) and g ( f(x) ). Determine whether the composition function is commutative (def. The same or equal) f(x) = 2x + 1 g(x) = 3 - x f(g(x)) = 2(g(x) + 1 = 2 ( 3 - x ) + 1 = 6 - 2x + 1 = 7 - 2x g(f(x)) = 3 - (f(x) = 3 - ( 2x + 1 ) = 3 - 2x - 1 = 2 - 2x These equations are not equal and therefore are not commutative Example 4 Suppose f(x) gives miles that can be driven in x hours and g(y) gives the gallons of gas u ...
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