Solving for f(x) when 2f(x) f(1/x) 6x^3/x

When working with algebraic functions, it is common to encounter equations that require you to manipulate and solve for a specific function. In this article, we will focus on solving for the function f(x) given the equation:

[2f(x) f(1/x) 6x^3/x]

Let's delve into the steps required to find f(x).

Method 1: Direct Comparison of Coefficients

Firstly, observe that the left-hand side (LHS) and right-hand side (RHS) of the given equation are 'comparison-friendly'. By comparing the coefficients of x and 1/x, we can deduce the form of f(x).

Given:

2f(x) f(1/x) 6x3/x

Let's compare the coefficients:

By setting the coefficient of x and 1/x on both sides equal, we can simplify the equation to:

f(x) 6x/2 3x

Method 2: Substitution and Manipulation

As an alternative approach, we can substitute x with 1/x in the original equation and then manipulate the equations to solve for f(x).

Step 1: Substituting x with 1/x

By substituting x with 1/x, we get:

2f(1/x) f(x) 6(1/x)3x

Which simplifies to:

2f(1/x) f(x) 6/x3x

Let's denote this as equation [2]:

2f(1/x) f(x) 6/x3x -- (2)

Step 2: Multiplying the Original Equation by 2

Multiplying the original equation by 2, we get:

4f(x) 2f(1/x) 12x^3/x

Let's denote this as equation [3]:

4f(x) 2f(1/x) 12x3/x -- (3)

Step 3: Subtracting Equation [2] from Equation [3]

Subtracting equation [2] from equation [3], we get:

2f(1/x) f(x) - 4f(x) - 2f(1/x) 6/x3x - 12x/3x

This simplifies to:

-3f(x) -9x

Therefore, solving for f(x), we get:

f(x) 3x

Conclusion: We have successfully determined that f(x) 3x.

Alternative Methods: Solving for f(x) Using Given Conditions

In the alternative method, we consider the scenario where x a or x 1/a. This leads to a system of equations that can be used to solve for f(x).

1. When x a: 2f(a) f(1/a) 6a3/a 2f(a) f(1/a) 6a2 -- (1)

2. When x 1/a: 2f(1/a) f(a) 6(1/a)3a 2f(1/a) f(a) 6/a3a 2f(1/a) f(a) 6/a2 -- (2)

Multiplying equation [1] by 2, we get: 4f(a) 2f(1/a) 12a2 -- (3)

Now, subtracting equation [2] from equation [3]: 4f(a) 2f(1/a) - 2f(1/a) - f(a) 12a2 - 6/a2 3f(a) 9a f(a) 3a

Conclusion: We have confirmed that f(x) 3x.