If $f(x)=x+8$ and $g(x)=-4 x-3$, find $(f-g)(x)$.
A. $(f-g)(x)=-3 x+5$
B. $(f-g)(x)=5 x+11$
C. $(f-g)(x)=-5 x-11$
D. $(f-g)(x)=-3 x-5

Subtract g ( x ) from f ( x ) : ( f − g ) ( x ) = f ( x ) − g ( x ) .
Substitute the given functions: ( f − g ) ( x ) = ( x + 8 ) − ( − 4 x − 3 ) .
Distribute the negative sign: ( f − g ) ( x ) = x + 8 + 4 x + 3 .
Combine like terms to find the result: ( f − g ) ( x ) = 5 x + 11 , so the answer is 5 x + 11 ​ .

Explanation

Understanding the problem We are given two functions, f ( x ) = x + 8 and g ( x ) = − 4 x − 3 , 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, substitute the given expressions for f ( x ) and g ( x ) : ( f − g ) ( x ) = ( x + 8 ) − ( − 4 x − 3 )

Distributing the negative sign Next, we distribute the negative sign to both terms inside the parentheses of g ( x ) : ( f − g ) ( x ) = x + 8 + 4 x + 3

Combining like terms Now, we combine like terms. We combine the x terms and the constant terms: ( f − g ) ( x ) = ( x + 4 x ) + ( 8 + 3 ) ( f − g ) ( x ) = 5 x + 11

Final Answer So, ( f − g ) ( x ) = 5 x + 11 . Comparing this to the given options, we see that it matches option B.


Examples
Understanding function operations like subtraction is crucial in many real-world applications. For instance, if f ( x ) represents the revenue from selling x items and g ( x ) represents the cost of producing x items, then ( f − g ) ( x ) gives the profit. If f ( x ) = x + 8 and g ( x ) = − 4 x − 3 , then the profit function is ( f − g ) ( x ) = 5 x + 11 . This means for every item sold, the profit increases by $5, and there's an initial profit of $11 even if no items are sold, which might represent a starting bonus or investment return.

Answered by GinnyAnswer | 2025-07-08