If [tex]f(x)=\frac{x}{3}-2[/tex] and [tex]g(x)=3 x^2+2 x-6[/tex], find [tex](f+g)(x)[/tex]
A. [tex]3 x^2+\frac{7}{3} x-8[/tex]
B. [tex]3 x^2+\frac{5}{3} x+4[/tex]
C. [tex]x^2+12[/tex]
D. [tex]3 x^2+\frac{2}{3} x-8[/tex]
Answer (2)
Add the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) = ( 3 x − 2 ) + ( 3 x 2 + 2 x − 6 ) .
Combine the x terms: 3 x + 2 x = 3 7 x .
Combine the constant terms: − 2 − 6 = − 8 .
The result is ( f + g ) ( x ) = 3 x 2 + 3 7 x − 8 , so the answer is 3 x 2 + 3 7 x − 8 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 3 x − 2 and g ( x ) = 3 x 2 + 2 x − 6 , and we want to find ( f + g ) ( x ) . This means we need to add the two functions together.
Adding the Functions To find ( f + g ) ( x ) , we add f ( x ) and g ( x ) :
( f + g ) ( x ) = f ( x ) + g ( x ) = ( 3 x − 2 ) + ( 3 x 2 + 2 x − 6 )
Combining Like Terms Now, we combine like terms. We have a 3 x 2 term, x terms, and constant terms. Combining the x terms, we have 3 x + 2 x . To add these, we need a common denominator, which is 3. So, 2 x = 3 6 x . Thus, 3 x + 3 6 x = 3 7 x .
Combining the constant terms, we have − 2 − 6 = − 8 .
So, we have: ( f + g ) ( x ) = 3 x 2 + 3 7 x − 8
Final Answer Therefore, ( f + g ) ( x ) = 3 x 2 + 3 7 x − 8 . Comparing this to the given options, we see that it matches option A.
Examples
In electrical engineering, if you have two voltage sources represented by functions f(x) and g(x), the total voltage in a series circuit would be (f+g)(x). Understanding how to combine these functions allows engineers to analyze and design circuits effectively.
The sum of the functions f ( x ) and g ( x ) is given by ( f + g ) ( x ) = 3 x 2 + 3 7 x − 8 . Therefore, the answer is option A. This represents a combined polynomial involving both quadratic and linear terms, plus a constant.
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