If [tex]f(x)=x^4-x^3+x^2[/tex] and [tex]g(x)=-x^2[/tex], where [tex]x \neq 0[/tex], what is [tex](f / g)(x)[/tex]?
A. [tex]x^2-x+1[/tex]
B. [tex]x^2+x+1[/tex]
C. [tex]-x^2+x-1[/tex]
D. [tex]-x^2-x-1[/tex]
Answer (1)
Divide f ( x ) by g ( x ) to find ( f / g ) ( x ) : ( f / g ) ( x ) = − x 2 x 4 − x 3 + x 2 .
Divide each term in the numerator by − x 2 : − x 2 x 4 − − x 2 x 3 + − x 2 x 2 .
Simplify each term: − x 2 + x − 1 .
The result is: − x 2 + x − 1 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x 4 − x 3 + x 2 and g ( x ) = − x 2 , and we want to find ( f / g ) ( x ) , which is the same as g ( x ) f ( x ) .
Setting up the Division To find ( f / g ) ( x ) , we need to divide f ( x ) by g ( x ) : ( f / g ) ( x ) = g ( x ) f ( x ) = − x 2 x 4 − x 3 + x 2 .
Dividing Each Term Now, we simplify the expression by dividing each term in the numerator by − x 2 : ( f / g ) ( x ) = − x 2 x 4 − − x 2 x 3 + − x 2 x 2 .
Simplifying the Expression Performing the division for each term, we get: ( f / g ) ( x ) = − x 2 + x − 1.
Final Answer Therefore, ( f / g ) ( x ) = − x 2 + x − 1 .
Examples
Understanding how to divide functions is crucial in many areas, such as analyzing the behavior of complex systems. For example, if f ( x ) represents the total cost of producing x items and g ( x ) represents the number of items produced, then ( f / g ) ( x ) gives the average cost per item. This concept is widely used in economics and business to understand cost structures and optimize production processes. By simplifying the ratio of two functions, we can gain insights into the underlying relationships and make informed decisions.
