For $f(x)=2 x+1$ and $g(x)=x^2-7$, find $(f-g)(x)$.
A. $-x^2+2 x+8$
B. $2 x^2-15$
C. $-x^2+2 x-6$
D. $x^2-2 x-8$
Answer (2)
Subtract g ( x ) from f ( x ) : ( f − g ) ( x ) = f ( x ) − g ( x ) .
Substitute the given expressions: ( f − g ) ( x ) = ( 2 x + 1 ) − ( x 2 − 7 ) .
Simplify the expression: ( f − g ) ( x ) = 2 x + 1 − x 2 + 7 = − x 2 + 2 x + 8 .
The final answer is: − x 2 + 2 x + 8 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 2 x + 1 and g ( x ) = x 2 − 7 , and we want to find ( f − g ) ( x ) . This means we need to subtract the function g ( x ) from the function f ( x ) .
Setting up the Subtraction To find ( f − g ) ( x ) , we subtract g ( x ) from f ( x ) : ( f − g ) ( x ) = f ( x ) − g ( x ) Now, we substitute the given expressions for f ( x ) and g ( x ) : ( f − g ) ( x ) = ( 2 x + 1 ) − ( x 2 − 7 )
Simplifying the Expression Next, we simplify the expression by distributing the negative sign and combining like terms: ( f − g ) ( x ) = 2 x + 1 − x 2 + 7 Rearranging the terms to match the standard polynomial form, we get: ( f − g ) ( x ) = − x 2 + 2 x + 1 + 7 ( f − g ) ( x ) = − x 2 + 2 x + 8
Comparing with Options Finally, we compare our result with the given options. Our result is − x 2 + 2 x + 8 , which matches option A.
Examples
Understanding function operations like subtraction is useful in many real-world scenarios. For example, if f ( x ) represents the revenue from selling x items and g ( x ) represents the cost of producing x items, then ( f − g ) ( x ) represents the profit. By analyzing the profit function, businesses can make informed decisions about production levels and pricing strategies. In this case, if the revenue function is f ( x ) = 2 x + 1 and the cost function is g ( x ) = x 2 − 7 , then the profit function is ( f − g ) ( x ) = − x 2 + 2 x + 8 .
The function (f-g)(x) is found by subtracting g(x) from f(x). The result is (f-g)(x) = -x^2 + 2x + 8, which matches Option A. Therefore, the final answer is -x^2 + 2x + 8.
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