For f(x)=√x and g(x)=x-6, find the following.
a. (f∘g)(x)
b. (g∘f)(x)
c. (f∘g)(10)
a. What is (f∘g)(x)?
(f∘g)(x)=(Simplify your answer.)
b. What is (g∘f)(x)?
(g∘f)(x)=(Simplify your answer.)
c. What is (f∘g)(10)?
(f∘g)(10)=(Simplify your answer.)
Answer (2)
Find ( f ∘ g ) ( x ) by substituting g ( x ) into f ( x ) : ( f ∘ g ) ( x ) = x − 6 .
Find ( g ∘ f ) ( x ) by substituting f ( x ) into g ( x ) : ( g ∘ f ) ( x ) = x − 6 .
Evaluate ( f ∘ g ) ( 10 ) by substituting x = 10 into ( f ∘ g ) ( x ) : ( f ∘ g ) ( 10 ) = 2 .
The final answers are ( f ∘ g ) ( x ) = x − 6 , ( g ∘ f ) ( x ) = x − 6 , and ( f ∘ g ) ( 10 ) = 2 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x and g ( x ) = x − 6 . We need to find the composite functions ( f ∘ g ) ( x ) and ( g ∘ f ) ( x ) , and then evaluate ( f ∘ g ) ( 10 ) .
Finding (f o g)(x) First, let's find ( f ∘ g ) ( x ) , which means f ( g ( x )) . We substitute g ( x ) into f ( x ) : ( f ∘ g ) ( x ) = f ( g ( x )) = f ( x − 6 ) = x − 6 So, ( f ∘ g ) ( x ) = x − 6 .
Finding (g o f)(x) Next, let's find ( g ∘ f ) ( x ) , which means g ( f ( x )) . We substitute f ( x ) into g ( x ) : ( g ∘ f ) ( x ) = g ( f ( x )) = g ( x ) = x − 6 So, ( g ∘ f ) ( x ) = x − 6 .
Evaluating (f o g)(10) Now, let's evaluate ( f ∘ g ) ( 10 ) . We substitute x = 10 into ( f ∘ g ) ( x ) = x − 6 : ( f ∘ g ) ( 10 ) = 10 − 6 = 4 = 2 So, ( f ∘ g ) ( 10 ) = 2 .
Conclusion Therefore, we have found the composite functions and evaluated the required expression.
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 discount and g ( x ) = 1.08 x represents the sales tax, then ( g ∘ f ) ( x ) = g ( f ( x )) = 1.08 ( 0.9 x ) = 0.972 x represents the final price after both the discount and the sales tax are applied. This shows how composite functions can model sequential operations.
We found that ( f ∘ g ) ( x ) = x − 6 and ( g ∘ f ) ( x ) = x − 6 . Additionally, evaluating ( f ∘ g ) ( 10 ) gives us a result of 2. These calculations show how to combine functions effectively using composition.
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