If $f(x)=2 x-1$ and $g(x)=x^2-3 x-2$, find $(f+g)(x)$
A. $(f+g)(x)=x-3$
B. $(f+g)(x)=x^2+5 x+3$
C. $(f+g)(x)=-x-3$
D. $(f+g)(x)=x^2-x-3$

Add the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) .
Substitute the expressions: ( f + g ) ( x ) = ( 2 x − 1 ) + ( x 2 − 3 x − 2 ) .
Combine like terms: ( f + g ) ( x ) = x 2 − x − 3 .
The final answer is x 2 − x − 3 ​ .

Explanation

Understanding the Problem We are given two functions, f ( x ) = 2 x − 1 and g ( x ) = x 2 − 3 x − 2 , and we want to find their sum, ( f + g ) ( x ) . This means we need to add the two functions together.

Adding the Functions To find ( f + g ) ( x ) , we add the expressions for f ( x ) and g ( x ) : ( f + g ) ( x ) = f ( x ) + g ( x ) ( f + g ) ( x ) = ( 2 x − 1 ) + ( x 2 − 3 x − 2 ) Now, we combine like terms.

Simplifying the Expression We have the expression ( 2 x − 1 ) + ( x 2 − 3 x − 2 ) . Let's group the like terms together: ( f + g ) ( x ) = x 2 + ( 2 x − 3 x ) + ( − 1 − 2 ) Now, we simplify the terms: ( f + g ) ( x ) = x 2 − x − 3

Final Answer The sum of the functions f ( x ) and g ( x ) is ( f + g ) ( x ) = x 2 − x − 3 . This corresponds to option D.


Examples
Understanding how to combine functions is essential in many real-world applications. For instance, if you're tracking the cost and revenue of a product, you might have a cost function, C ( x ) , and a revenue function, R ( x ) , where x is the number of units sold. The profit function, P ( x ) , is then the difference between the revenue and cost functions, P ( x ) = R ( x ) − C ( x ) . By combining these functions, you can analyze the profitability of the product at different sales levels. Similarly, in physics, you might combine position and velocity functions to understand the motion of an object.

Answered by GinnyAnswer | 2025-07-08