If [tex]f(x)=\frac{x}{2}-3[/tex] and [tex]g(x)=3 x^2+x-6[/tex], find [tex](f+g)(x)[/tex].
A. [tex]4 x^2-9[/tex]
B. [tex]3 x^2+\frac{3}{2} x-9[/tex]
C. [tex]3 x^2+\frac{3}{2} x-3[/tex]
D. [tex]3 x^2-\frac{1}{2} x-3[/tex]
Answer (2)
Add the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) = ( f r a c x 2 − 3 ) + ( 3 x 2 + x − 6 ) .
Combine like terms: ( f + g ) ( x ) = 3 x 2 + ( f r a c x 2 + x ) + ( − 3 − 6 ) .
Simplify the expression: ( f + g ) ( x ) = 3 x 2 + f r a c 3 2 x − 9 .
The final answer is: 3 x 2 + 2 3 x − 9 .
Explanation
Understanding the problem We are given two functions, f ( x ) = f r a c x 2 − 3 and g ( x ) = 3 x 2 + x − 6 , and we want to find ( f + g ) ( x ) , which 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 ) = ( f r a c x 2 − 3 ) + ( 3 x 2 + x − 6 ) Now, we combine like terms.
Combining like terms We rearrange the terms to group like terms together: ( f + g ) ( x ) = 3 x 2 + ( f r a c x 2 + x ) + ( − 3 − 6 ) Now, we simplify the expression.
Simplifying the expression We have f r a c x 2 + x = f r a c x 2 + f r a c 2 x 2 = f r a c 3 x 2 and − 3 − 6 = − 9 . So, ( f + g ) ( x ) = 3 x 2 + f r a c 3 2 x − 9 Thus, ( f + g ) ( x ) = 3 x 2 + f r a c 3 2 x − 9 .
Final Answer Comparing our result with the given options, we see that it matches option B. Therefore, the correct answer is 3 x 2 + f r a c 3 2 x − 9 .
Examples
Understanding how to combine functions is useful in many real-world scenarios. For example, if you have a business, you might have a cost function, C ( x ) , that represents the cost of producing x items, and a revenue function, R ( x ) , that represents the revenue from selling x items. The profit function, P ( x ) , is the difference between the revenue and cost functions: P ( x ) = R ( x ) − C ( x ) . By understanding how to combine functions, you can analyze the profit of your business and make informed decisions.
When adding the functions f(x) and g(x), we combine like terms to find (f+g)(x) = 3x^2 + \frac{3}{2}x - 9. Therefore, the correct answer is option B: 3x^2 + \frac{3}{2}x - 9.
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