Question 10 (5 points)
Given that [tex]f(x)=4 x+5[/tex] and [tex]g(x)=x^2-2 x-2[/tex], find [tex](f+g)(x)[/tex].
A) [tex](f+g)(x)=x^2+6 x+3[/tex]
B) [tex](f+g)(x)=-x^2+6 x+7[/tex]
C) [tex](f+g)(x)=-x^2+2 x-7[/tex]
D) [tex](f+g)(x)=x^2+2 x+3[/tex]
Answer (1)
Add the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) .
Substitute the given expressions: ( f + g ) ( x ) = ( 4 x + 5 ) + ( x 2 − 2 x − 2 ) .
Combine like terms: ( f + g ) ( x ) = x 2 + ( 4 x − 2 x ) + ( 5 − 2 ) .
Simplify to find the final expression: ( f + g ) ( x ) = x 2 + 2 x + 3 . The answer is x 2 + 2 x + 3 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 4 x + 5 and g ( x ) = x 2 − 2 x − 2 . Our goal is to find the sum of these two functions, which is denoted as ( f + g ) ( x ) . This means we need to add the expressions for f ( x ) and g ( x ) and simplify the result.
Adding the Functions To find ( f + g ) ( x ) , we simply add the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) Substitute the given expressions for f ( x ) and g ( x ) :
( f + g ) ( x ) = ( 4 x + 5 ) + ( x 2 − 2 x − 2 ) Now, we combine like terms to simplify the expression.
Simplifying the Expression Combine the x 2 terms, the x terms, and the constant terms: ( f + g ) ( x ) = x 2 + ( 4 x − 2 x ) + ( 5 − 2 ) Simplify the expression: ( f + g ) ( x ) = x 2 + 2 x + 3
Final Answer The sum of the functions f ( x ) and g ( x ) is x 2 + 2 x + 3 . Therefore, ( f + g ) ( x ) = x 2 + 2 x + 3 .
Comparing this result with the given options, we see that it matches option D. So the answer is D) ( f + g ) ( x ) = x 2 + 2 x + 3
Examples
Understanding how to combine functions is useful in many real-world scenarios. For example, if you have a business where your revenue R ( x ) depends on the number of items sold x , and your cost C ( x ) also depends on the number of items sold, then your profit P ( x ) can be found by subtracting the cost function from the revenue function: P ( x ) = R ( x ) − C ( x ) . This is an example of combining functions to model a real-world situation.
