Understanding the Functional Equation Problem

The problem asks us to find the value of \(f(2)\) given a functional equation involving \(f(x)\) and \(f \left(\frac{1}{x}\right)\). The given equation is:

\[4f(x) - f \left(\frac{1}{x}\right)=\left(2x+\frac{1}{x}\right)\left(2x-\frac{1}{x}\right)\]

The right-hand side of the equation is in the form \((a+b)(a-b)\), which simplifies to \(a^2 - b^2\). Here, \(a = 2x\) and \(b = \frac{1}{x}\).

Simplifying the right-hand side:

\[\left(2x+\frac{1}{x}\right)\left(2x-\frac{1}{x}\right) = (2x)^2 - \left(\frac{1}{x}\right)^2 = 4x^2 - \frac{1}{x^2}\]

So, the given functional equation becomes:

\[4f(x) - f \left(\frac{1}{x}\right) = 4x^2 - \frac{1}{x^2} \quad (*)\label{eq:1}\]

Solving the System of Functional Equations

To find \(f(x)\), we can use a common technique for this type of functional equation: substitute \(x\) with \(\frac{1}{x}\) in the original equation. Let's substitute \(\frac{1}{x}\) for \(x\) in equation (*):

\[4f\left(\frac{1}{x}\right) - f\left(\frac{1}{\frac{1}{x}}\right) = 4\left(\frac{1}{x}\right)^2 - \frac{1}{\left(\frac{1}{x}\right)^2}\]

\[4f\left(\frac{1}{x}\right) - f(x) = \frac{4}{x^2} - x^2\]

Let's rearrange this new equation:

\[-f(x) + 4f\left(\frac{1}{x}\right) = \frac{4}{x^2} - x^2 \quad (**)\label{eq:2}\]

Now we have a system of two linear equations with two variables, \(f(x)\) and \(f\left(\frac{1}{x}\right)\):

1. \(4f(x) - f \left(\frac{1}{x}\right) = 4x^2 - \frac{1}{x^2}\)
2. \(-f(x) + 4f\left(\frac{1}{x}\right) = \frac{4}{x^2} - x^2\)

We can solve this system using elimination. Multiply equation (1) by 4:

\[4 \times (4f(x) - f \left(\frac{1}{x}\right)) = 4 \times \left(4x^2 - \frac{1}{x^2}\right)\]

\[16f(x) - 4f \left(\frac{1}{x}\right) = 16x^2 - \frac{4}{x^2} \quad (***)\label{eq:3}\]

Now, add equation (**) and equation (***):

\[(-f(x) + 4f\left(\frac{1}{x}\right)) + (16f(x) - 4f \left(\frac{1}{x}\right)) = \left(\frac{4}{x^2} - x^2\right) + \left(16x^2 - \frac{4}{x^2}\right)\]

The terms involving \(f\left(\frac{1}{x}\right)\) cancel out:

\[-f(x) + 16f(x) = \frac{4}{x^2} - x^2 + 16x^2 - \frac{4}{x^2}\]

\[15f(x) = (-x^2 + 16x^2) + \left(\frac{4}{x^2} - \frac{4}{x^2}\right)\]

\[15f(x) = 15x^2 + 0\]

\[15f(x) = 15x^2\]

Divide by 15 to find \(f(x)\):

\[f(x) = \frac{15x^2}{15}\]

\[f(x) = x^2\]

Calculating f(2)

We have found that the function is \(f(x) = x^2\). Now we need to find the value of \(f(2)\).

Substitute \(x = 2\) into the expression for \(f(x)\):

\[f(2) = (2)^2\]

\[f(2) = 4\]

Thus, \(f(2)\) is equal to 4.

Revision Table: Key Concepts

Additional Information: Functional Equation Types

Functional equations can take many forms. The one solved here is a linear functional equation because it involves linear combinations of \(f(x)\) and \(f(1/x)\).

• Cauchy's Functional Equations: Basic forms like \(f(x+y) = f(x) + f(y)\).
• Linear Functional Equations: Equations involving linear combinations of the function evaluated at different points or transformed inputs, like the one in this problem.
• Recurrence Relations: Define a sequence where each term is given as a function of preceding terms (e.g., Fibonacci sequence). These can be seen as functional equations over integers.

Solving functional equations often involves substitution, looking for specific values (like \(f(0)\) or \(f(1)\)), or exploring properties like linearity or continuity.