Problem 17 - Calculus One Again

Question

Find all functions \(f: \mathbb{R} \backslash\\{0\\} \rightarrow \mathbb{R}\) for which $$ f(x)+2 f\left(\frac{1}{x}\right)=x $$

Step-by-Step Solution

Answer

The function is \(f(x) = \frac{2 - x^2}{3x}\).

Step 1: Understand the Functional Equation

We need to find functions \(f(x)\) such that for all \(xeq 0\), the equation \(f(x) + 2f\left(\frac{1}{x}\right) = x\) holds. This involves expressing the function elements in terms of other input values.

Step 2: Substitute and Simplify

Consider the equation \(f(x) + 2f\left(\frac{1}{x}\right) = x\). If we let \(y = \frac{1}{x}\), then solving for \(x\) gives \(x = \frac{1}{y}\). Now substitute these into the equation. You get \(f\left(\frac{1}{y}\right) + 2f(y) = \frac{1}{y}\).

Step 3: Write System of Equations

Now, we have two equations: 1. \(f(x) + 2f\left(\frac{1}{x}\right) = x\) 2. \(f\left(\frac{1}{x}\right) + 2f(x) = \frac{1}{x}\)These need to be solved simultaneously to find \(f(x)\).

Step 4: Solve System for f(x) and f(1/x)

Taking these equations: - \(f(x) + 2f\left(\frac{1}{x}\right) = x\) - \(2f(x) + f\left(\frac{1}{x}\right) = \frac{1}{x}\)Multiply the second equation by 2 to aid elimination: - \(4f(x) + 2f\left(\frac{1}{x}\right) = \frac{2}{x}\)Now subtract the first equation from this scaled version: - \(3f(x) = \frac{2}{x} - x\)Thus, \(f(x) = \frac{2}{3x} - \frac{x}{3}\).

Step 5: Verify Solution

To ensure consistency, substitute back into both original equations. Insert \(f(x) = \frac{2 - x^2}{3x}\) into \(f(\frac{1}{x})\), then verify that both equations are satisfied, confirming that this \(f(x)\) meets the equation criteria.

Key Concepts

Substitution Method

The substitution method is a powerful tool in solving functional equations. It involves replacing variables with expressions to simplify and solve equations. In the original exercise, we first need to understand the given functional equation: \(f(x) + 2f\left(\frac{1}{x}\right) = x\). Let's say we substitute \(y = \frac{1}{x}\). By applying this substitution, we transform \(f\left(\frac{1}{y}\right) + 2f(y) = \frac{1}{y}\). The goal of substitution is to express the function in different terms that can help simplify and ultimately solve the problem. This strategy helps us in developing relationships between various forms of the function, laying down the groundwork for further manipulation. By systematically replacing and rearranging, substitution serves as an essential step in many algebraic methods for tackling functional equations.

System of Equations

A system of equations is a set of multiple equations that work together under a common solution. Solving a system means finding values that satisfy all the equations simultaneously.In our exercise, after using substitution, we crafted two main equations:

• \(f(x) + 2f\left(\frac{1}{x}\right) = x\)
• \(f\left(\frac{1}{x}\right) + 2f(x) = \frac{1}{x}\)

Solving the system requires leveraging both equations in such a way that each complements the other. This means finding one common function that satisfies both. Techniques such as adding, subtracting, or combining these equations in different manners allow uncovering of the precise function mapping required.By establishing equations that interrelate the function in a systematic manner, it becomes more straightforward to unravel the unknowns involved. Thus, understanding and solving a system of equations is pivotal in achieving the correct function.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve equations. It requires a good grasp of algebra rules and properties, like distribution, combining like terms, and inverses.In the exercise, after setting up the system of equations, algebraic manipulation becomes the key to isolating \(f(x)\). We see this when multiplying the second equation by 2 and subsequently subtracting the first equation:

• Start: \(2\big(f(x) + f\left(\frac{1}{x}\right)\big) = \frac{2}{x}\)
• Transformation: \(4f(x) + 2f\left(\frac{1}{x}\right) = \frac{2}{x}\)
• Subtract: \(4f(x) + 2f\left(\frac{1}{x}\right) - (f(x) + 2f\left(\frac{1}{x}\right)) = \frac{2}{x} - x\)

This procedure results in simplifying down to \(3f(x) = \frac{2}{x} - x\), leading directly to finding \(f(x) = \frac{2}{3x} - \frac{x}{3}\).Manipulation is thus a detailed yet necessary process that makes it feasible to extract an explicit solution from more convoluted equations. Recognizing effective manipulation sequences often makes the difference in rapidly and correctly solving equations.