For [tex]$f(x)=2 x-3$[/tex] and [tex]$g(x)=5 x^2-4$[/tex], find the following:
a. [tex]$(f \circ g)(x)$[/tex]
b. [tex]$(g \circ f)(x)$[/tex]
c. [tex]$(f \circ g)(2)$[/tex]

a. What is [tex]$(f \circ g)(x)$[/tex]?
[tex]$(f \circ g)(x)=[/tex] $\square$ (Simplify your answer.)

Find ( f ∘ g ) ( x ) by substituting g ( x ) into f ( x ) and simplifying: ( f ∘ g ) ( x ) = 10 x 2 − 11 .
Find ( g ∘ f ) ( x ) by substituting f ( x ) into g ( x ) and simplifying: ( g ∘ f ) ( x ) = 20 x 2 − 60 x + 41 .
Evaluate ( f ∘ g ) ( 2 ) by substituting x = 2 into the expression for ( f ∘ g ) ( x ) : ( f ∘ g ) ( 2 ) = 29 .
The final answers are ( f ∘ g ) ( x ) = 10 x 2 − 11 , ( g ∘ f ) ( x ) = 20 x 2 − 60 x + 41 , and ( f ∘ g ) ( 2 ) = 29 ​ .

Explanation

Understanding the Problem We are given two functions, f ( x ) = 2 x − 3 and g ( x ) = 5 x 2 − 4 . Our goal is to find the composite functions ( f ∘ g ) ( x ) and ( g ∘ f ) ( x ) , and then evaluate ( f ∘ g ) ( 2 ) .

Finding ( f ∘ g ) ( x ) To find ( f ∘ g ) ( x ) , we need to substitute g ( x ) into f ( x ) . This means we replace x in f ( x ) with the entire function g ( x ) . So, we have: ( f ∘ g ) ( x ) = f ( g ( x )) = f ( 5 x 2 − 4 ) Now, we substitute 5 x 2 − 4 into f ( x ) = 2 x − 3 :
f ( 5 x 2 − 4 ) = 2 ( 5 x 2 − 4 ) − 3

Simplifying ( f ∘ g ) ( x ) Next, we simplify the expression for ( f ∘ g ) ( x ) :
2 ( 5 x 2 − 4 ) − 3 = 10 x 2 − 8 − 3 = 10 x 2 − 11 So, ( f ∘ g ) ( x ) = 10 x 2 − 11 .

Finding ( g ∘ f ) ( x ) To find ( g ∘ f ) ( x ) , we need to substitute f ( x ) into g ( x ) . This means we replace x in g ( x ) with the entire function f ( x ) . So, we have: ( g ∘ f ) ( x ) = g ( f ( x )) = g ( 2 x − 3 ) Now, we substitute 2 x − 3 into g ( x ) = 5 x 2 − 4 :
g ( 2 x − 3 ) = 5 ( 2 x − 3 ) 2 − 4

Simplifying ( g ∘ f ) ( x ) Next, we simplify the expression for ( g ∘ f ) ( x ) :
5 ( 2 x − 3 ) 2 − 4 = 5 ( 4 x 2 − 12 x + 9 ) − 4 = 20 x 2 − 60 x + 45 − 4 = 20 x 2 − 60 x + 41 So, ( g ∘ f ) ( x ) = 20 x 2 − 60 x + 41 .

Finding ( f ∘ g ) ( 2 ) Finally, we need to find ( f ∘ g ) ( 2 ) . We already found that ( f ∘ g ) ( x ) = 10 x 2 − 11 . So, we substitute x = 2 into this expression: ( f ∘ g ) ( 2 ) = 10 ( 2 ) 2 − 11 = 10 ( 4 ) − 11 = 40 − 11 = 29 So, ( f ∘ g ) ( 2 ) = 29 .

Final Answer Therefore, we have: ( f ∘ g ) ( x ) = 10 x 2 − 11 ( g ∘ f ) ( x ) = 20 x 2 − 60 x + 41 ( f ∘ g ) ( 2 ) = 29


Examples
Composite functions are useful in many real-world scenarios. For example, consider a store that offers a discount of 10% on all items, and then applies a sales tax of 8%. If f ( x ) = 0.9 x represents the price after the discount and g ( x ) = 1.08 x represents the price after sales tax, then ( g ∘ f ) ( x ) represents the final price you pay for an item. Understanding composite functions helps you analyze the combined effect of multiple operations or transformations.

Answered by GinnyAnswer | 2025-07-07

The composite functions are ( f ∘ g ) ( x ) = 10 x 2 − 11 and ( g ∘ f ) ( x ) = 20 x 2 − 60 x + 41 . Evaluating ( f ∘ g ) ( 2 ) gives a result of 29.
;

Answered by Anonymous | 2025-08-26