If [tex]f(x)=3^x+10x[/tex] and [tex]g(x)=4x-2[/tex], find [tex](f+g)(x)[/tex]
A. [tex]$17x-2$[/tex]
B. [tex]$3^x+14x-2$[/tex]
C. [tex]$3^x+6x+2$[/tex]
D. [tex]$3^x-6x+2$[/tex]
Answer (2)
Define the sum of functions: ( f + g ) ( x ) = f ( x ) + g ( x ) .
Substitute the given expressions: ( f + g ) ( x ) = ( 3 x + 10 x ) + ( 4 x − 2 ) .
Combine like terms: ( f + g ) ( x ) = 3 x + ( 10 x + 4 x ) − 2 .
Simplify the expression: ( f + g ) ( x ) = 3 x + 14 x − 2 . The answer is 3 x + 14 x − 2 .
Explanation
Understanding the problem We are given two functions, f ( x ) = 3 x + 10 x and g ( x ) = 4 x − 2 . Our goal is to find the sum of these two functions, which is denoted as ( f + g ) ( x ) .
Defining the sum of functions The sum of two functions is defined as ( f + g ) ( x ) = f ( x ) + g ( x ) . This means we need to add the expressions for f ( x ) and g ( x ) together.
Substituting the expressions Now, let's substitute the given expressions for f ( x ) and g ( x ) into the equation: ( f + g ) ( x ) = ( 3 x + 10 x ) + ( 4 x − 2 )
Combining like terms Next, we combine like terms. In this case, we combine the terms with x :
( f + g ) ( x ) = 3 x + ( 10 x + 4 x ) − 2 ( f + g ) ( x ) = 3 x + 14 x − 2
Final Answer Therefore, the sum of the two functions is ( f + g ) ( x ) = 3 x + 14 x − 2 . Looking at the multiple-choice options, we see that this matches option B.
Examples
Understanding how to combine functions is useful in many real-world scenarios. For example, if a company's revenue is modeled by one function and its costs by another, combining these functions can give you a profit function. If f ( x ) represents the revenue from selling x items and g ( x ) represents the cost of producing x items, then ( f − g ) ( x ) would represent the profit. This allows businesses to analyze their financial performance and make informed decisions.
The sum of the functions is found using the formula ( f + g ) ( x ) = f ( x ) + g ( x ) . After substituting the expressions for f ( x ) and g ( x ) and simplifying, we find that ( f + g ) ( x ) = 3 x + 14 x − 2 , which corresponds to option B. Therefore, the answer is option B, 3 x + 14 x − 2 .
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