Key Concepts and Formulas

• Functional Equations: Equations involving unknown functions. A common technique for equations with $x$ and $1/x$ is symmetric substitution.
• Symmetric Substitution: Replacing $x$ with $1/x$ in a functional equation to generate a new equation.
• System of Linear Equations: Two or more equations with the same variables that can be solved simultaneously.

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Step-by-Step Solution

Step 1: Understand the Given Information and Objective We are given the following conditions:

1. $a + \alpha = 1$
2. $b + \beta = 2$
3. The functional equation: $af(x) + \alpha f\left( {{1 \over x}} \right) = bx + {\beta \over x} \quad (*)$ Our objective is to find the value of the expression $\frac{f(x) + f\left( {{1 \over x}} \right)}{x + {1 \over x}}$.

Step 2: Apply Symmetric Substitution to the Functional Equation The functional equation contains terms involving $f(x)$ and $f(1/x)$. To create a system of equations that can help us solve for $f(x) + f(1/x)$, we will substitute $x$ with $1/x$ in the given functional equation $(*)$. This is a standard technique for functional equations exhibiting symmetry. Replacing $x$ with $1/x$ in $(*)$: $a f\left( {{1 \over x}} \right) + \alpha f\left( {{1 \over {1/x}}} \right) = b\left( {{1 \over x}} \right) + {\beta \over {1/x}}$ Simplifying the arguments of the function and the terms on the right-hand side: $a f\left( {{1 \over x}} \right) + \alpha f(x) = {b \over x} + \beta x \quad (**)$

Step 3: Formulate a System of Linear Equations We now have two equations involving $f(x)$ and $f(1/x)$: Equation $(*)$: $af(x) + \alpha f\left( {{1 \over x}} \right) = bx + {\beta \over x}$ Equation $(**)$: $\alpha f(x) + a f\left( {{1 \over x}} \right) = \beta x + {b \over x}$ (Rearranging the RHS of (**) to match the order in (*))

Step 4: Add the Two Equations to Isolate the Sum of Functions Our aim is to find an expression for $f(x) + f(1/x)$. By adding the two equations in the system, we can group the terms involving $f(x)$ and $f(1/x)$ together. Adding Equation $(*)$ and Equation $(**)$: $(af(x) + \alpha f\left( {{1 \over x}} \right)) + (\alpha f(x) + a f\left( {{1 \over x}} \right)) = \left( bx + {\beta \over x} \right) + \left( \beta x + {b \over x} \right)$ Group the terms on the LHS by $f(x)$ and $f(1/x)$, and on the RHS by $x$ and $1/x$: $(a + \alpha)f(x) + (\alpha + a)f\left( {{1 \over x}} \right) = (b + \beta)x + \left( \frac{\beta}{x} + \frac{b}{x} \right)$ Factor out the common coefficients $(a + \alpha)$ on the LHS and $(b + \beta)$ on the RHS: $(a + \alpha)\left[ f(x) + f\left( {{1 \over x}} \right) \right] = (b + \beta)x + \frac{(\beta + b)}{x}$ $(a + \alpha)\left[ f(x) + f\left( {{1 \over x}} \right) \right] = (b + \beta)\left( x + {1 \over x} \right)$

Step 5: Rearrange to Find the Desired Expression We need to find the value of $\frac{f(x) + f\left( {{1 \over x}} \right)}{x + {1 \over x}}$. To achieve this, we divide both sides of the equation from Step 4 by $(x + 1/x)$ and $(a + \alpha)$, assuming $x \ne 0$ and $a + \alpha \ne 0$. $\frac{f(x) + f\left( {{1 \over x}} \right)}{x + {1 \over x}} = \frac{b + \beta}{a + \alpha}$

Step 6: Substitute the Given Values Now, we use the given conditions: $a + \alpha = 1$ and $b + \beta = 2$. Substitute these values into the expression obtained in Step 5: $\frac{f(x) + f\left( {{1 \over x}} \right)}{x + {1 \over x}} = \frac{2}{1}$ $\frac{f(x) + f\left( {{1 \over x}} \right)}{x + {1 \over x}} = 2$

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Common Mistakes & Tips

• Algebraic Errors: Be meticulous with algebraic manipulations, especially when simplifying terms like $1/(1/x)$ and when combining fractions.
• Systematic Approach: Always ensure that when you substitute $x$ with $1/x$, you consistently apply it to all terms in the equation.
• Recognizing the Goal: Keep the target expression $\frac{f(x) + f\left( {{1 \over x}} \right)}{x + {1 \over x}}$ in mind. This helps guide the steps, particularly when deciding whether to add or subtract the equations in the system.

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Summary

The problem involves a functional equation with a symmetric structure ($x$ and $1/x$). The key strategy is to use symmetric substitution, replacing $x$ with $1/x$ to generate a second equation. These two equations form a system that can be solved for the desired expression. By adding the two equations and then substituting the given relations ($a + \alpha = 1$ and $b + \beta = 2$), we directly obtain the value of $\frac{f(x) + f\left( {{1 \over x}} \right)}{x + {1 \over x}}$.

The final answer is $\boxed{2}$.