Solving the Division of Functions: f(x)/g(x) Given f(x) 2x^2 and g(x) 2x^3

In algebra, the division of functions is a fundamental operation that often arises when simplifying expressions or solving equations. Given two functions, ( f(x) 2x^2 ) and ( g(x) 2x^3 ), the process of finding ( frac{f(x)}{g(x)} ) involves simplifying the quotient of these two expressions. This guide provides a step-by-step solution and explores the underlying concepts of algebraic functions and their division.

Understanding the Given Functions

In the given problem, we have two functions:

Function ( f(x) )

The function ( f(x) 2x^2 ) is a quadratic function, where the variable ( x ) is squared and multiplied by 2. This means that for any input value ( x ), the function ( f(x) ) yields ( 2x^2 ).

Function ( g(x) )

The function ( g(x) 2x^3 ) is a cubic function, where the variable ( x ) is cubed and multiplied by 2. For any input value ( x ), the function ( g(x) ) results in ( 2x^3 ).

Solving for ( frac{f(x)}{g(x)} )

Given the functions ( f(x) 2x^2 ) and ( g(x) 2x^3 ), we need to find the quotient of these two functions, denoted as ( frac{f(x)}{g(x)} ).

To solve this, we simply divide the function ( f(x) ) by the function ( g(x) ) as follows:

( frac{f(x)}{g(x)} frac{2x^2}{2x^3} )

Let's simplify this expression step-by-step.

Step 1: Factor the Numerator and Denominator

We start by expressing both the numerator and the denominator in their factored forms:

Numerator: ( 2x^2 )

Denominator: ( 2x^3 )

Step 2: Simplify the Expression

To simplify the expression ( frac{2x^2}{2x^3} ), we can cancel out the common factor of 2 in the numerator and the denominator:

( frac{2x^2}{2x^3} frac{2 cdot x^2}{2 cdot x^3} frac{x^2}{x^3} )

Step 3: Simplify the Variables

Next, we simplify the remaining variables. Recall that ( frac{x^a}{x^b} x^{a-b} ) where ( a ) and ( b ) are the exponents:

( frac{x^2}{x^3} x^{2-3} x^{-1} )

Finally, we can rewrite ( x^{-1} ) as ( frac{1}{x} ) because ( x^{-1} frac{1}{x^1} frac{1}{x} ).

Therefore, the simplified form of ( frac{f(x)}{g(x)} ) is:

( frac{f(x)}{g(x)} x^{-1} frac{1}{x} )

Conceptual Understanding

Understanding the concepts of function division is crucial in algebra. It involves dividing the numerator and the denominator by their common factors and simplifying the resulting expression. This process helps in solving more complex algebraic equations and simplifying expressions to their most basic form.

Alternative Approach: Using Specific Values

To further illustrate the concept, we can use a specific example. Let's consider ( y 9 ) and ( x 3 ). We can then find the value of ( frac{y}{x} ) as follows:

( frac{y}{x} frac{9}{3} 3 )

This method helps in verifying the correctness of the algebraic manipulation and provides a practical understanding of how the result can be obtained for specific values of the variables.

Conclusion

In conclusion, the division of functions ( f(x) 2x^2 ) and ( g(x) 2x^3 ) results in the simplified expression ( frac{1}{x} ). This method of function division is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems. Practicing these steps and exploring specific values can enhance your understanding and problem-solving skills in algebra.