If [tex]f(x)=3^x+10 x[/tex] and [tex]g(x)=5 x-3[/tex], find [tex](f-g)(x)[/tex].
A. [tex]18 x-3[/tex]
B. [tex]3^x-5 x+3[/tex]
C. [tex]3^x+15 x-3[/tex]
D. [tex]3^x+5 x+3[/tex]
Answer (2)
Find the difference between the functions: ( f − g ) ( x ) = f ( x ) − g ( x ) .
Substitute the given functions: ( f − g ) ( x ) = ( 3 x + 10 x ) − ( 5 x − 3 ) .
Simplify the expression by combining like terms: ( f − g ) ( x ) = 3 x + 10 x − 5 x + 3 = 3 x + 5 x + 3 .
The final answer is 3 x + 5 x + 3 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 3 x + 10 x and g ( x ) = 5 x − 3 . Our goal is to find the expression for ( f − g ) ( x ) , which means we need to subtract the function g ( x ) from the function f ( x ) .
Subtracting the Functions To find ( f − g ) ( x ) , we subtract g ( x ) from f ( x ) : ( f − g ) ( x ) = f ( x ) − g ( x ) Substituting the given functions, we have: ( f − g ) ( x ) = ( 3 x + 10 x ) − ( 5 x − 3 ) Now, we simplify the expression by removing the parentheses and combining like terms.
Simplifying the Expression Distribute the negative sign to each term inside the second parentheses: ( f − g ) ( x ) = 3 x + 10 x − 5 x + 3 Combine the like terms (the terms with x ): ( f − g ) ( x ) = 3 x + ( 10 x − 5 x ) + 3 ( f − g ) ( x ) = 3 x + 5 x + 3
Final Answer The resulting expression for ( f − g ) ( x ) is 3 x + 5 x + 3 . Comparing this with the given options, we see that it matches option D. Therefore, the correct answer is D.
Examples
Understanding function operations like subtraction is crucial in many real-world applications. For instance, consider a scenario where 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) made from producing and selling x items. By analyzing this difference, businesses can make informed decisions about production levels and pricing strategies to maximize their profit.
The difference between the functions f ( x ) and g ( x ) is given by ( f − g ) ( x ) = 3 x + 5 x + 3 . This matches option D from the provided choices. Therefore, the correct answer is D.
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