For [tex]f(x)=\frac{8}{x}[/tex] and [tex]g(x)=\frac{8}{x}[/tex] find
a. [tex](f \circ g)(x)[/tex];
b. [tex](g \circ f)(x)[/tex];
c. [tex](f \circ g)(5)[/tex]
a. [tex](f \circ g)(x)=[/tex] (Simplify your answer.)
Answer (1)
Find the composite function ( f ∘ g ) ( x ) by substituting g ( x ) into f ( x ) , resulting in f ( g ( x )) = f ( x 8 ) = x 8 8 .
Simplify the expression x 8 8 to obtain ( f ∘ g ) ( x ) = x .
Evaluate ( f ∘ g ) ( 5 ) by substituting x = 5 into the simplified expression, yielding ( f ∘ g ) ( 5 ) = 5 .
The final answer is x .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x 8 and g ( x ) = x 8 . We need to find the composition of these functions, ( f ∘ g ) ( x ) and ( g ∘ f ) ( x ) , and then evaluate ( f ∘ g ) ( 5 ) .
Finding (f \circ g)(x) To find ( f ∘ g ) ( x ) , we need to find f ( g ( x )) . This means we substitute g ( x ) into f ( x ) . Since g ( x ) = x 8 , we have f ( g ( x )) = f ( x 8 ) .
Simplifying (f \circ g)(x) Now, we substitute x 8 into f ( x ) = x 8 . So, f ( x 8 ) = x 8 8 . To simplify this expression, we can multiply the numerator and denominator by x , which gives us 8 8 x = x . Therefore, ( f ∘ g ) ( x ) = x .
Finding (g \circ f)(x) To find ( g ∘ f ) ( x ) , we need to find g ( f ( x )) . This means we substitute f ( x ) into g ( x ) . Since f ( x ) = x 8 , we have g ( f ( x )) = g ( x 8 ) .
Simplifying (g \circ f)(x) Now, we substitute x 8 into g ( x ) = x 8 . So, g ( x 8 ) = x 8 8 . To simplify this expression, we can multiply the numerator and denominator by x , which gives us 8 8 x = x . Therefore, ( g ∘ f ) ( x ) = x .
Evaluating (f \circ g)(5) To find ( f ∘ g ) ( 5 ) , we substitute x = 5 into the expression we found for ( f ∘ g ) ( x ) , which is x . Therefore, ( f ∘ g ) ( 5 ) = 5 .
Final Answer In summary: a. ( f ∘ g ) ( x ) = x b. ( g ∘ f ) ( x ) = x c. ( f ∘ g ) ( 5 ) = 5
Examples
Function composition is a fundamental concept in mathematics with numerous real-world applications. For instance, consider a store offering a discount where f ( x ) represents a 20% discount on an item's price x , and g ( x ) represents a $5 coupon applied after the discount. The composite function ( f ∘ g ) ( x ) would calculate the final price after applying the discount first and then the coupon. Understanding function composition helps in modeling sequential operations and analyzing their combined effect, which is crucial in fields like economics, computer science, and engineering.
