Given that [tex]$f(x)=x^2+5 x$[/tex] and [tex]$g(x)=-5 x-5$[/tex], determine each of the following. Make sure to fully simplify your answer.
(a) [tex]$(f \circ g)(x)=[/tex] $\square$
(b) [tex]$(g \circ f)(x)=[/tex] $\square$
(c) [tex]$(f \circ f)(x)=[/tex] $\square$
(d) [tex]$(g \circ g)(x)=[/tex] $\square$

Calculate ( f ∘ g ) ( x ) by substituting g ( x ) into f ( x ) and simplifying: ( f ∘ g ) ( x ) = f ( − 5 x − 5 ) = 25 x 2 + 25 x .
Calculate ( g ∘ f ) ( x ) by substituting f ( x ) into g ( x ) and simplifying: ( g ∘ f ) ( x ) = g ( x 2 + 5 x ) = − 5 x 2 − 25 x − 5 .
Calculate ( f ∘ f ) ( x ) by substituting f ( x ) into f ( x ) and simplifying: ( f ∘ f ) ( x ) = f ( x 2 + 5 x ) = x 4 + 10 x 3 + 30 x 2 + 25 x .
Calculate ( g ∘ g ) ( x ) by substituting g ( x ) into g ( x ) and simplifying: ( g ∘ g ) ( x ) = g ( − 5 x − 5 ) = 25 x + 20 .

Final Answers: ( f ∘ g ) ( x ) = 25 x 2 + 25 x ​ ( g ∘ f ) ( x ) = − 5 x 2 − 25 x − 5 ​ ( f ∘ f ) ( x ) = x 4 + 10 x 3 + 30 x 2 + 25 x ​ $\boxed{(g \circ g)(x) = 25x + 20}
Explanation

Understanding the Problem We are given two functions, f ( x ) = x 2 + 5 x and g ( x ) = − 5 x − 5 . Our goal is to find the composite functions ( f ∘ g ) ( x ) , ( g ∘ f ) ( x ) , ( f ∘ f ) ( x ) , and ( g ∘ g ) ( x ) . Remember that ( f ∘ g ) ( x ) means f ( g ( x )) , and similarly for the other compositions.

Calculating (f o g)(x) (a) To find ( f ∘ g ) ( x ) , we need to substitute g ( x ) into f ( x ) . So, we have


f ( g ( x )) = f ( − 5 x − 5 ) = ( − 5 x − 5 ) 2 + 5 ( − 5 x − 5 ) .
Expanding this, we get:
( − 5 x − 5 ) 2 = ( − 5 x ) 2 + 2 ( − 5 x ) ( − 5 ) + ( − 5 ) 2 = 25 x 2 + 50 x + 25 .
And 5 ( − 5 x − 5 ) = − 25 x − 25 .
So, f ( g ( x )) = 25 x 2 + 50 x + 25 − 25 x − 25 = 25 x 2 + 25 x .

Calculating (g o f)(x) (b) To find ( g ∘ f ) ( x ) , we need to substitute f ( x ) into g ( x ) . So, we have

g ( f ( x )) = g ( x 2 + 5 x ) = − 5 ( x 2 + 5 x ) − 5 .
Expanding this, we get:
− 5 ( x 2 + 5 x ) − 5 = − 5 x 2 − 25 x − 5 .

Calculating (f o f)(x) (c) To find ( f ∘ f ) ( x ) , we need to substitute f ( x ) into f ( x ) . So, we have

f ( f ( x )) = f ( x 2 + 5 x ) = ( x 2 + 5 x ) 2 + 5 ( x 2 + 5 x ) .
Expanding this, we get:
( x 2 + 5 x ) 2 = ( x 2 ) 2 + 2 ( x 2 ) ( 5 x ) + ( 5 x ) 2 = x 4 + 10 x 3 + 25 x 2 .
And 5 ( x 2 + 5 x ) = 5 x 2 + 25 x .
So, f ( f ( x )) = x 4 + 10 x 3 + 25 x 2 + 5 x 2 + 25 x = x 4 + 10 x 3 + 30 x 2 + 25 x .

Calculating (g o g)(x) (d) To find ( g ∘ g ) ( x ) , we need to substitute g ( x ) into g ( x ) . So, we have

g ( g ( x )) = g ( − 5 x − 5 ) = − 5 ( − 5 x − 5 ) − 5 .
Expanding this, we get:
− 5 ( − 5 x − 5 ) − 5 = 25 x + 25 − 5 = 25 x + 20 .

Final Answer Therefore, the composite functions are:

(a) ( f ∘ g ) ( x ) = 25 x 2 + 25 x
(b) ( g ∘ f ) ( x ) = − 5 x 2 − 25 x − 5
(c) ( f ∘ f ) ( x ) = x 4 + 10 x 3 + 30 x 2 + 25 x
(d) ( g ∘ g ) ( x ) = 25 x + 20
Examples
Understanding composite functions is crucial in many areas of mathematics and real-world applications. For instance, consider a store offering a discount where f ( x ) represents a 20% off coupon on an item's price x , and g ( x ) represents a $10 rebate on the discounted price. The composite function ( f ∘ g ) ( x ) would calculate the final price after applying the rebate to the discounted price, while ( g ∘ f ) ( x ) would calculate the final price after applying the discount to the price after the rebate. This concept is also used in physics to describe transformations and in computer science to chain operations.

Answered by GinnyAnswer | 2025-07-07

The compositions of the functions are: (f \circ g)(x) = 25x^2 + 25x, (g \circ f)(x) = -5x^2 - 25x - 5, (f \circ f)(x) = x^4 + 10x^3 + 30x^2 + 25x, and (g \circ g)(x) = 25x + 20.
;

Answered by Anonymous | 2025-07-25