Problem 149 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Given \(a f(x)+b f\left(\frac{1}{x}\right)=\frac{1}{x}-5, x \neq 0, a \neq b\). Find \(f(x)\).

Step-by-Step Solution

Answer

The function \(f(x)\) is defined as \(f(x) = f\left(\frac{1}{x}\right)\).

Step 1 - Substitution

Substitute \(u = \frac{1}{x}\). This gives us two equations: \(a f(x)+b f(u)=u-5\) and \(a f(u)+b f(x)=x-5\). These equations look less intimidating and easier to solve. Here \(u\) stands for \(\frac{1}{x}\).

Step 2 - Find Equal Terms

Setting these equations equal to each other and considering the term that contains \(f(x)\), the equation acquired is \(a f(x) - a f(u) + b f(u) - b f(x) = 0\), this is due to the fact that \(u-5 = x-5\).

Step 3 - Rearrange and Simplify

Reorganizing the terms and simplifying will yield \((a-b) f(x) = (a-b) f(u)\).

Step 4 - Simplify

Since the problem states that \(a \neq b\), we can divide both sides by \((a-b)\), simplifying to \(f(x) = f(u)\).

Step 5 - Substitute u back

Finally, remember that \(u = \frac{1}{x}\). This gives us \(f(x) = f\left(\frac{1}{x}\right)\) as the equation for the function \(f(x)\).

Key Concepts

Substitution Method

When dealing with functional equations, the substitution method is a powerful tool that makes the process of finding solutions more manageable. The objective here is to transform the given problem into an equivalent one that is easier to solve.

The initial equation given is:

• \(a f(x)+b f\left(\frac{1}{x}\right)=\frac{1}{x}-5\)

To simplify this, a substitution technique can be employed:

• Set \(u = \frac{1}{x}\)

This substitution creates a new variable \(u\), which transforms the function and assists in simplifying the equation. Two transformed equations are derived:

• \(a f(x) + b f(u) = u - 5\)
• \(a f(u) + b f(x) = x - 5\)

By introducing \(u\), a symmetry in the problem is revealed, enabling further steps to be taken more effectively.

Equation Rearrangement

Rearranging equations is a fundamental step in problem-solving as it relies on the principle of equivalence, maintaining balance in mathematical expressions while transforming their form.

For the given functional equations:

• \(a f(x) + b f(u) = u - 5\)
• \(a f(u) + b f(x) = x - 5\)

The goal is to focus on rearranging terms in each equation to isolate or simplify one particular element, like \(f(x)\). By setting terms in these equations equal to each other:

• \(a f(x) - a f(u) + b f(u) - b f(x) = 0\)

This step reveals the symmetry and interdependence between the terms, setting the stage for simplification, a crucial preparation for solving the equation.

Mathematics Problem Solving

Mathematics problem-solving involves a systematic approach to analyzing and solving equations, especially functional ones. The technique used here includes exploring symmetric properties in the equations and linking back to previously made substitutions and rearrangements.

A critical step involves simplifying the rearranged equation to reveal:

• \((a-b) f(x) = (a-b) f(u)\)

Given the condition \(a eq b\), it allows us to divide both sides by \(a-b\), leading to:

• \(f(x) = f(u)\)

This solution shows the functional symmetry, indicating that \(f(x)\) and \(f(u)\) are equivalent across the given domain, demonstrating the importance of structure in equations.

Symmetry in Functions

Understanding symmetry in functions is a central consideration in solving equations, particularly when substitution and rearrangement lead to equivalent expressions.

In our example, after simplifying and utilizing the equations again, symmetry emerges:

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

This shows that \(f(x)\) remains the same if \(x\) is replaced by its reciprocal \(\frac{1}{x}\). Considering symmetry assists in predicting the behavior of the function across different inputs.

This symmetry is crucial because it not only helps in solving the equation but also gives insights into the nature of the function \(f(x)\) itself. Recognizing such patterns makes functional equation problems easier to approach and understand.