For f(x)=√x and g(x) = x - 1, find the following.
a. (f ∘ g)(x)
b. (g ∘ f)(x)
c. (f ∘ g)(5)
What is (f ∘ g)(x)?
Answer (1)
Find ( f ∘ g ) ( x ) by substituting g ( x ) into f ( x ) : ( f ∘ g ) ( x ) = f ( g ( x )) = f ( x − 1 ) = x − 1 .
Find ( g ∘ f ) ( x ) by substituting f ( x ) into g ( x ) : ( g ∘ f ) ( x ) = g ( f ( x )) = g ( x ) = x − 1 .
Find ( f ∘ g ) ( 5 ) by substituting x = 5 into ( f ∘ g ) ( x ) = x − 1 : ( f ∘ g ) ( 5 ) = 5 − 1 = 4 = 2 .
The final answer is ( f ∘ g ) ( x ) = x − 1 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x and g ( x ) = x − 1 . We need to find the composite functions ( f ∘ g ) ( x ) and ( g ∘ f ) ( x ) , and then evaluate ( f ∘ g ) ( 5 ) .
Finding (f o g)(x) First, let's find ( f ∘ g ) ( x ) . This means we need to substitute g ( x ) into f ( x ) . So, we have ( f ∘ g ) ( x ) = f ( g ( x )) = f ( x − 1 ) = x − 1 .
Finding (g o f)(x) Next, let's find ( g ∘ f ) ( x ) . This means we need to substitute f ( x ) into g ( x ) . So, we have ( g ∘ f ) ( x ) = g ( f ( x )) = g ( x ) = x − 1.
Evaluating (f o g)(5) Now, let's evaluate ( f ∘ g ) ( 5 ) . We substitute x = 5 into the expression we found for ( f ∘ g ) ( x ) :
( f ∘ g ) ( 5 ) = 5 − 1 = 4 = 2.
Final Answer Therefore, we have found that ( f ∘ g ) ( x ) = x − 1 , ( g ∘ f ) ( x ) = x − 1 , and ( f ∘ g ) ( 5 ) = 2 .
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 tax, then ( g ∘ f ) ( x ) represents the final price you pay. Understanding composite functions helps in analyzing such situations.
