If $f(x)=x^2+1$ and $g(x)=x-4$, which value is equivalent to $(f \circ g)(10)$?
A. 37
B. 97
C. 126
D. 606
Answer (1)
First, find the value of g ( 10 ) by substituting 10 into g ( x ) = x − 4 , which gives g ( 10 ) = 6 .
Next, substitute the result into f ( x ) = x 2 + 1 to find f ( g ( 10 )) = f ( 6 ) .
Calculate f ( 6 ) = 6 2 + 1 = 36 + 1 = 37 .
Therefore, ( f ∘ g ) ( 10 ) = 37 , so the final answer is 37 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x 2 + 1 and g ( x ) = x − 4 . We need to find the value of the composite function ( f ∘ g ) ( 10 ) , which means we need to find f ( g ( 10 )) .
Calculating g(10) First, we need to find the value of g ( 10 ) . We substitute x = 10 into the function g ( x ) = x − 4 :
g ( 10 ) = 10 − 4 = 6
Calculating f(g(10)) Now that we have g ( 10 ) = 6 , we need to find f ( g ( 10 )) , which is f ( 6 ) . We substitute x = 6 into the function f ( x ) = x 2 + 1 :
f ( 6 ) = 6 2 + 1 = 36 + 1 = 37
Final Answer Therefore, ( f ∘ g ) ( 10 ) = f ( g ( 10 )) = f ( 6 ) = 37 .
Examples
Composite functions are used in many real-world applications. For example, in manufacturing, a company might have a function that calculates the cost of producing x items, c ( x ) , and another function that calculates the revenue from selling x items, r ( x ) . The profit function, p ( x ) , can be expressed as a composite function p ( x ) = r ( x ) − c ( x ) . Evaluating composite functions helps the company understand their profit at different production levels.
