If [tex]f(x)=x^2+1[/tex] and [tex]g(x)=x-4[/tex], which value is equivalent to [tex](f \circ g)(10)[/tex] ?
A. 37
B. 97
C. 126
D. 606
Answer (2)
First, find the value of g ( 10 ) : g ( 10 ) = 10 − 4 = 6 .
Then, substitute the result into f ( x ) : f ( 6 ) = 6 2 + 1 = 36 + 1 = 37 .
Thus, ( f ∘ g ) ( 10 ) = 37 .
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 ) . This means we first evaluate g ( 10 ) , and then we use that result as the input for the function f ( x ) .
Evaluating g(10) First, let's find g ( 10 ) . We substitute x = 10 into the expression for g ( x ) : g ( 10 ) = 10 − 4 = 6
Evaluating f(g(10)) Now that we have g ( 10 ) = 6 , we can find f ( g ( 10 )) = f ( 6 ) . We substitute x = 6 into the expression for f ( x ) : f ( 6 ) = 6 2 + 1 = 36 + 1 = 37
Final Answer Therefore, ( f ∘ g ) ( 10 ) = 37 .
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 5% to the discounted price. If d ( x ) = 0.9 x represents the discounted price and t ( x ) = 1.05 x represents the price after tax, then the composite function t ( d ( x )) represents the final price you pay for an item. Evaluating composite functions helps in understanding the combined effect of multiple operations or transformations.
To find ( f ∘ g ) ( 10 ) , first evaluate g ( 10 ) = 6 , then substitute this into f : f ( 6 ) = 37 . The final answer is A. 37 .
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