For [tex]$f(x)=6 x-5$[/tex] and [tex]$g(x)=3 x^2-4$[/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.)
b. What is [tex]$(g \circ f)(x)$[/tex]?
[tex]$(g \circ f)(x)=[/tex] $\square$ (Simplify your answer.)
c. What is [tex]$(f \circ g)(2)$[/tex]?
[tex]$(f \circ g)(2)=[/tex] $\square$ (Simplify your answer.)
Answer (1)
Find ( f ∘ g ) ( x ) by substituting g ( x ) into f ( x ) and simplifying: ( f ∘ g ) ( x ) = 18 x 2 − 29 .
Find ( g ∘ f ) ( x ) by substituting f ( x ) into g ( x ) and simplifying: ( g ∘ f ) ( x ) = 108 x 2 − 180 x + 71 .
Evaluate ( f ∘ g ) ( 2 ) by substituting x = 2 into ( f ∘ g ) ( x ) : ( f ∘ g ) ( 2 ) = 43 .
The final answers are ( f ∘ g ) ( x ) = 18 x 2 − 29 , ( g ∘ f ) ( x ) = 108 x 2 − 180 x + 71 , and ( f ∘ g ) ( 2 ) = 43 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 6 x − 5 and g ( x ) = 3 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 ( 3 x 2 − 4 ) = 6 ( 3 x 2 − 4 ) − 5
Simplifying ( f ∘ g ) ( x ) Now, we simplify the expression: ( f ∘ g ) ( x ) = 6 ( 3 x 2 − 4 ) − 5 = 18 x 2 − 24 − 5 = 18 x 2 − 29
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 ( 6 x − 5 ) = 3 ( 6 x − 5 ) 2 − 4
Simplifying ( g ∘ f ) ( x ) Now, we simplify the expression: ( g ∘ f ) ( x ) = 3 ( 6 x − 5 ) 2 − 4 = 3 ( 36 x 2 − 60 x + 25 ) − 4 = 108 x 2 − 180 x + 75 − 4 = 108 x 2 − 180 x + 71
Finding ( f ∘ g ) ( 2 ) To find ( f ∘ g ) ( 2 ) , we substitute x = 2 into the expression we found for ( f ∘ g ) ( x ) :
( f ∘ g ) ( 2 ) = 18 ( 2 ) 2 − 29 = 18 ( 4 ) − 29 = 72 − 29 = 43
Final Answer Therefore, we have: ( f ∘ g ) ( x ) = 18 x 2 − 29 ( g ∘ f ) ( x ) = 108 x 2 − 180 x + 71 ( f ∘ g ) ( 2 ) = 43
Examples
Composite functions are useful in many real-world scenarios. For example, consider a store that marks up the price of an item by 20%, and then offers a 10% discount on all items. If f ( x ) = 1.2 x represents the markup and g ( x ) = 0.9 x represents the discount, then ( g ∘ f ) ( x ) represents the final price of an item after both the markup and the discount are applied. Understanding composite functions helps businesses analyze pricing strategies and understand the combined effect of multiple operations.
