If [tex]f(x)=-x^2+6 x-1[/tex] and [tex]g(x)=3 x^2-4 x-1[/tex], find [tex](f-g)(x)[/tex]
A. [tex](f-g)(x)=4 x^2-10 x[/tex]
B. [tex](f-g)(x)=-4 x^2-2 x[/tex]
C. [tex](f-g)(x)=2 x^2+2 x-2[/tex]
D. [tex](f-g)(x)=-4 x^2+10 x[/tex]
Answer (1)
Subtract g ( x ) from f ( x ) : ( f − g ) ( x ) = f ( x ) − g ( x ) .
Distribute the negative sign: ( f − g ) ( x ) = − x 2 + 6 x − 1 − ( 3 x 2 − 4 x − 1 ) = − x 2 + 6 x − 1 − 3 x 2 + 4 x + 1 .
Combine like terms: ( f − g ) ( x ) = ( − x 2 − 3 x 2 ) + ( 6 x + 4 x ) + ( − 1 + 1 ) .
Simplify: ( f − g ) ( x ) = − 4 x 2 + 10 x . The answer is − 4 x 2 + 10 x .
Explanation
Understanding the Problem We are given two functions, f ( x ) = − x 2 + 6 x − 1 and g ( x ) = 3 x 2 − 4 x − 1 , and we want to find ( f − g ) ( x ) , which means f ( x ) − g ( x ) .
Subtracting the Functions To find ( f − g ) ( x ) , we subtract g ( x ) from f ( x ) : ( f − g ) ( x ) = f ( x ) − g ( x ) = ( − x 2 + 6 x − 1 ) − ( 3 x 2 − 4 x − 1 ) We need to distribute the negative sign to each term in g ( x ) .
Distributing the Negative Sign Distributing the negative sign, we get: ( f − g ) ( x ) = − x 2 + 6 x − 1 − 3 x 2 + 4 x + 1 Now, we combine like terms.
Combining Like Terms Combining the x 2 terms, we have − x 2 − 3 x 2 = − 4 x 2 .
Combining the x terms, we have 6 x + 4 x = 10 x .
Combining the constant terms, we have − 1 + 1 = 0 .
So, ( f − g ) ( x ) = − 4 x 2 + 10 x + 0 = − 4 x 2 + 10 x .
Final Answer Therefore, ( f − g ) ( x ) = − 4 x 2 + 10 x . Comparing this to the given options, we see that it matches option D.
Examples
Understanding function subtraction is useful in many real-world scenarios. For example, if you have a revenue function f ( x ) and a cost function g ( x ) , where x is the number of units sold, then the profit function is given by ( f − g ) ( x ) , which represents the revenue minus the cost. By finding the difference between these functions, you can determine the profit for a given number of units sold. This helps in making informed business decisions.
