Animated Solution for Mathematics - Functions: If $a+\alpha =1$, $b+\beta =2$ and $af(x)+\alpha f(\frac{1}{x})=bx+\frac{\beta }{x}$, $x\ne 0$, then the value of the expression $\frac{f(x)+f(\frac{1}{x})}{x+\frac{1}{x}}$ is

Imagine standing before a complex functional equation. It looks intimidating, doesn't it? You see $f(x)$ and $f(\frac{1}{x})$ dancing together, and your instinct might be to panic, to try and isolate $f(x)$ using brute force.

But stop. Take a breath. In the world of JEE Advanced, functional equations are not problems to be 'solved' in the traditional sense; they are puzzles to be 'unlocked' through symmetry.

Today, we are going to dismantle this problem not with brute force, but with the elegance of a master strategist.

The Power of the Swap

We are given the equation:

Look at the arguments. We have $x$ and $\frac{1}{x}$. This is a classic signature of a reciprocal symmetry.

Whenever you see $x$ and $\frac{1}{x}$ in a functional equation, your brain should immediately trigger a specific reflex: the substitution $x\to \frac{1}{x}$. This transformation swaps the positions of the function terms and creates a mirror image of your original equation.

Replacing $x$ with $\frac{1}{x}$ in our original equation gives us:

Rearranging this slightly, we get:

Now, we have transformed a single, confusing equation into a structured system of two equations. This is the first step in turning chaos into order.

Adding for Clarity

Now, let's look at our two equations side-by-side:

Equation (i): $af(x)+\alpha f\left(\frac{1}{x}\right)=bx+\frac{\beta }{x}$

Equation (ii): $\alpha f(x)+af\left(\frac{1}{x}\right)=\beta x+\frac{b}{x}$

What happens when we add them? This is where the magic happens. On the left-hand side, we get:

By adding them, we have factored out $(a+\alpha )$. On the right-hand side, we get:

We have effectively collapsed the complexity of the function into a simple linear relationship:

The Beauty of Constants

Look at the constants provided in the problem: $a+\alpha =1$ and $b+\beta =2$. This is not a coincidence; it is the architect of the problem handing you the key to the final door.

Substitute these values into our equation. The expression becomes:

Suddenly, the entire functional complexity vanishes. Dividing both sides by $(x+\frac{1}{x})$, we arrive at the target expression:

The final answer is 2. It is clean, it is elegant, and it is absolute.

This problem teaches us that in JEE Advanced, the path to the solution is rarely through the longest calculation, but through the deepest observation. When you see symmetry, embrace it. Use it. Let it guide you to the answer.