For [tex]f(x) = 3x[/tex] and [tex]g(x) = x + 3[/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] (Simplify your answer.)
Answer (2)
Find ( f c i rc g ) ( x ) by substituting g ( x ) into f ( x ) : ( f c i rc g ) ( x ) = f ( g ( x )) = f ( x + 3 ) = 3 ( x + 3 ) = 3 x + 9 .
Find ( g c i rc f ) ( x ) by substituting f ( x ) into g ( x ) : ( g c i rc f ) ( x ) = g ( f ( x )) = g ( 3 x ) = 3 x + 3 .
Find ( f c i rc g ) ( 2 ) by substituting x = 2 into ( f c i rc g ) ( x ) : ( f c i rc g ) ( 2 ) = 3 ( 2 ) + 9 = 15 .
The final answers are ( f c i rc g ) ( x ) = b o x e d 3 x + 9 , ( g c i rc f ) ( x ) = 3 x + 3 , and ( f c i rc g ) ( 2 ) = 15 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 3 x and g ( x ) = x + 3 . We need to find the composite functions ( f c i rc g ) ( x ) and ( g c i rc f ) ( x ) , and then evaluate ( f c i rc g ) ( 2 ) .
Finding (f o g)(x) To find ( f c i rc g ) ( x ) , we substitute g ( x ) into f ( x ) . This means we replace x in f ( x ) with g ( x ) , which is x + 3 . So, we have
( f c i rc g ) ( x ) = f ( g ( x )) = f ( x + 3 ) = 3 ( x + 3 )
Now, we simplify the expression:
3 ( x + 3 ) = 3 x + 9
So, ( f c i rc g ) ( x ) = 3 x + 9 .
Finding (g o f)(x) To find ( g c i rc f ) ( x ) , we substitute f ( x ) into g ( x ) . This means we replace x in g ( x ) with f ( x ) , which is 3 x . So, we have
( g c i rc f ) ( x ) = g ( f ( x )) = g ( 3 x ) = 3 x + 3
So, ( g c i rc f ) ( x ) = 3 x + 3 .
Finding (f o g)(2) To find ( f c i rc g ) ( 2 ) , we substitute x = 2 into the expression we obtained for ( f c i rc g ) ( x ) , which is 3 x + 9 . So, we have
( f c i rc g ) ( 2 ) = 3 ( 2 ) + 9 = 6 + 9 = 15
So, ( f c i rc g ) ( 2 ) = 15 .
Final Answer Therefore, a. ( f c i rc g ) ( x ) = 3 x + 9 b. ( g c i rc f ) ( x ) = 3 x + 3 c. $(f circ g)(2) = 15
Examples
Composite functions are used in various real-life 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 discount and g ( x ) = 1.08 x represents the sales tax, then ( g c i rc f ) ( x ) represents the final price after both the discount and the sales tax are applied. Understanding composite functions helps in analyzing such sequential operations.
We found that ( f ∘ g ) ( x ) = 3 x + 9 and ( g ∘ f ) ( x ) = 3 x + 3 . Also, the value of ( f ∘ g ) ( 2 ) is 15. These calculations demonstrate how to find composite functions and evaluate them at a specific point.
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