Given that [tex]f(x)=x^2-4x[/tex] and [tex]g(x)=-3x+9[/tex], determine each of the following. You may simplify your answer.

(a) [tex](f \circ g)(x)=[/tex]
(b) [tex](g \circ f)(x)=[/tex]
(c) [tex](f \circ f)(x)=[/tex]
(d) [tex](g \circ g)(x)=[/tex]

Calculate ( f c i rc g ) ( x ) by substituting g ( x ) into f ( x ) and simplifying: ( f c i rc g ) ( x ) = 9 x 2 − 42 x + 45 .
Calculate ( g c i rc f ) ( x ) by substituting f ( x ) into g ( x ) and simplifying: ( g c i rc f ) ( x ) = − 3 x 2 + 12 x + 9 .
Calculate ( f c i rc f ) ( x ) by substituting f ( x ) into f ( x ) and simplifying: ( f c i rc f ) ( x ) = x 4 − 8 x 3 + 12 x 2 + 16 x .
Calculate ( g c i rc g ) ( x ) by substituting g ( x ) into g ( x ) and simplifying: ( g c i rc g ) ( x ) = 9 x − 18 .
The final answers are: ( f c i rc g ) ( x ) = b o x e d 9 x 2 − 42 x + 45 , ( g c i rc f ) ( x ) = b o x e d − 3 x 2 + 12 x + 9 , ( f c i rc f ) ( x ) = b o x e d x 4 − 8 x 3 + 12 x 2 + 16 x , ( g c i rc g ) ( x ) = b o x e d 9 x − 18 .

Explanation

Understanding the Problem We are given two functions, f ( x ) = x 2 − 4 x and g ( x ) = − 3 x + 9 . We need to find the composite functions ( f ∘ g ) ( x ) , ( g ∘ f ) ( x ) , ( f ∘ f ) ( x ) , and ( g ∘ g ) ( x ) . This involves substituting one function into another and simplifying the resulting expression.

Calculating (f o g)(x) (a) To find ( f ∘ g ) ( x ) , we need to compute f ( g ( x )) . This means we substitute g ( x ) into f ( x ) . So, we have: f ( g ( x )) = f ( − 3 x + 9 ) = ( − 3 x + 9 ) 2 − 4 ( − 3 x + 9 ) Expanding and simplifying: ( − 3 x + 9 ) 2 = ( − 3 x ) 2 + 2 ( − 3 x ) ( 9 ) + 9 2 = 9 x 2 − 54 x + 81 − 4 ( − 3 x + 9 ) = 12 x − 36 Therefore, f ( g ( x )) = 9 x 2 − 54 x + 81 + 12 x − 36 = 9 x 2 − 42 x + 45

Calculating (g o f)(x) (b) To find ( g ∘ f ) ( x ) , we need to compute g ( f ( x )) . This means we substitute f ( x ) into g ( x ) . So, we have: g ( f ( x )) = g ( x 2 − 4 x ) = − 3 ( x 2 − 4 x ) + 9 Expanding and simplifying: g ( f ( x )) = − 3 x 2 + 12 x + 9

Calculating (f o f)(x) (c) To find ( f ∘ f ) ( x ) , we need to compute f ( f ( x )) . This means we substitute f ( x ) into f ( x ) . So, we have: f ( f ( x )) = f ( x 2 − 4 x ) = ( x 2 − 4 x ) 2 − 4 ( x 2 − 4 x ) Expanding and simplifying: ( x 2 − 4 x ) 2 = ( x 2 − 4 x ) ( x 2 − 4 x ) = x 4 − 4 x 3 − 4 x 3 + 16 x 2 = x 4 − 8 x 3 + 16 x 2 − 4 ( x 2 − 4 x ) = − 4 x 2 + 16 x Therefore, f ( f ( x )) = x 4 − 8 x 3 + 16 x 2 − 4 x 2 + 16 x = x 4 − 8 x 3 + 12 x 2 + 16 x

Calculating (g o g)(x) (d) To find ( g ∘ g ) ( x ) , we need to compute g ( g ( x )) . This means we substitute g ( x ) into g ( x ) . So, we have: g ( g ( x )) = g ( − 3 x + 9 ) = − 3 ( − 3 x + 9 ) + 9 Expanding and simplifying: g ( g ( x )) = 9 x − 27 + 9 = 9 x − 18

Final Answer Therefore, the composite functions are: (a) ( f ∘ g ) ( x ) = 9 x 2 − 42 x + 45 (b) ( g ∘ f ) ( x ) = − 3 x 2 + 12 x + 9 (c) ( f ∘ f ) ( x ) = x 4 − 8 x 3 + 12 x 2 + 16 x (d) ( g ∘ g ) ( x ) = 9 x − 18


Examples
Composite functions are useful in many real-world scenarios. For example, if a store is offering a discount of 20% on all items, and you also have a coupon for 10 o ff , yo u c an u seco m p os i t e f u n c t i o n s t o d e t er min e t h e f ina lp r i ce . L e t f(x) = 0.8x re p rese n tt h e 20 g(x) = x - 10 re p rese n tt h eco u p o n . T h e n (f \circ g)(x) = f(g(x)) = 0.8(x - 10)$ represents the final price after applying both the discount and the coupon. Understanding composite functions helps in optimizing such calculations.

Answered by GinnyAnswer | 2025-07-07

The composite functions of the given functions are: (a) ( f ⌢ g ) ( x ) = 9 x 2 − 42 x + 45 , (b) ( g ⌢ f ) ( x ) = − 3 x 2 + 12 x + 9 , (c) ( f ⌢ f ) ( x ) = x 4 − 8 x 3 + 12 x 2 + 16 x , and (d) ( g ⌢ g ) ( x ) = 9 x − 18 .
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Answered by Anonymous | 2025-07-08