If [tex]$f(x)=3 x-1$[/tex] and [tex]$g(x)=x+2$[/tex], find [tex]$(f-g)(x)$[/tex]
A. [tex]$4 x+1$[/tex]
B. [tex]$2 x-3$[/tex]
C. [tex]$3-2 x$[/tex]
D. [tex]$2 x-1$[/tex]
Answer (2)
Subtract g ( x ) from f ( x ) : ( f − g ) ( x ) = f ( x ) − g ( x ) .
Substitute the given expressions: ( f − g ) ( x ) = ( 3 x − 1 ) − ( x + 2 ) .
Simplify the expression: ( f − g ) ( x ) = 3 x − 1 − x − 2 .
Combine like terms to find the result: ( f − g ) ( x ) = 2 x − 3 , so the answer is 2 x − 3 .
Explanation
Understanding the problem We are given two functions, f ( x ) = 3 x − 1 and g ( x ) = x + 2 . Our goal is to find the function ( f − g ) ( x ) , which means we need to subtract g ( x ) from f ( x ) .
Subtracting the functions To find ( f − g ) ( x ) , we subtract the function g ( x ) from the function f ( x ) : ( f − g ) ( x ) = f ( x ) − g ( x )
Substituting the expressions Now, we substitute the given expressions for f ( x ) and g ( x ) :
( f − g ) ( x ) = ( 3 x − 1 ) − ( x + 2 )
Simplifying the expression Next, we simplify the expression by removing the parentheses and combining like terms: ( f − g ) ( x ) = 3 x − 1 − x − 2 ( f − g ) ( x ) = ( 3 x − x ) + ( − 1 − 2 ) ( f − g ) ( x ) = 2 x − 3
Final Answer Therefore, ( f − g ) ( x ) = 2 x − 3 . Looking at the given options, we see that this matches option B.
Examples
Understanding function operations like ( f − g ) ( x ) is useful in many real-world scenarios. For example, imagine f ( x ) represents the total cost of producing x items, and g ( x ) represents the revenue generated from selling x items. Then, ( f − g ) ( x ) would represent the profit (or loss) from producing and selling x items. By analyzing this new function, you can determine the break-even point, maximize profit, or minimize losses.
We calculated ( f − g ) ( x ) by subtracting g ( x ) from f ( x ) , resulting in the expression 2 x − 3 . This matches the option B provided. Thus, the answer is 2 x − 3 .
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