Key Concepts and Formulas

• Functional Equations: Equations involving an unknown function and its arguments. Solving them often involves strategic substitution of values for variables.
• Properties of Functions: Understanding how to manipulate function arguments and use given conditions (like $f(0)=1$) is crucial.
• Algebraic Manipulation: Basic algebraic skills are needed to simplify expressions and solve for unknowns.

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Step-by-Step Solution

Step 1: Understanding the Problem and Strategy We are given a functional equation $f(x - y) = f(x)f(y) - f(a - x)f(a + y)$ and the condition $f(0) = 1$. We need to find the value of $f(2a - x)$. The strategy will be to first find the value of $f(a)$ by substituting specific values into the functional equation, and then use this information to evaluate $f(2a - x)$.

Step 2: Determining the Value of $f(a)$ To find $f(a)$, we will substitute $x=0$ and $y=0$ into the given functional equation. $f(x - y) = f(x)f(y) - f(a - x)f(a + y)$ Substituting $x=0$ and $y=0$: $f(0 - 0) = f(0)f(0) - f(a - 0)f(a + 0)$ $f(0) = [f(0)]^2 - [f(a)]^2$ We are given that $f(0) = 1$. Substituting this value: $1 = (1)^2 - [f(a)]^2$ $1 = 1 - [f(a)]^2$ Rearranging the terms to solve for $f(a)$: $[f(a)]^2 = 1 - 1$ $[f(a)]^2 = 0$ Therefore, we conclude that: $f(a) = 0$

Step 3: Expressing $f(2a-x)$ in a Usable Form Our goal is to find $f(2a - x)$. We need to express the argument $2a - x$ as a difference of two terms, say $X - Y$, so that we can apply the functional equation $f(X - Y) = f(X)f(Y) - f(a - X)f(a + Y)$. We can rewrite $2a - x$ as $a - (x - a)$. Let $X = a$ and $Y = x - a$. Then, $f(2a - x) = f(a - (x - a))$.

Step 4: Applying the Functional Equation to Find $f(2a-x)$ Now, we substitute $X=a$ and $Y=x-a$ into the functional equation: $f(X - Y) = f(X)f(Y) - f(a - X)f(a + Y)$ $f(a - (x - a)) = f(a)f(x - a) - f(a - a)f(a + (x - a))$ Let's simplify the terms on the right-hand side:

• $f(a)$ is known to be $0$ from Step 2.
• $f(a - a) = f(0)$, which is given as $1$.
• $f(a + (x - a)) = f(a + x - a) = f(x)$.

Substituting these back into the equation: $f(2a - x) = f(a)f(x - a) - f(0)f(x)$ $f(2a - x) = (0) \cdot f(x - a) - (1) \cdot f(x)$ $f(2a - x) = 0 - f(x)$ $f(2a - x) = -f(x)$

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Common Mistakes & Tips

• Incorrect Substitution: Ensure that when substituting values for $x$ and $y$, all occurrences on both sides of the functional equation are correctly replaced.
• Forgetting $f(0)=1$ or $f(a)=0$: These derived and given values are critical for simplifying the expression. Do not overlook them.
• Algebraic Errors: Be meticulous with algebraic manipulations, especially when dealing with signs and combining terms.

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Summary

The problem was solved by first utilizing the given condition $f(0)=1$ and the functional equation to determine that $f(a)=0$. Subsequently, the argument $2a-x$ was strategically expressed as a difference $a-(x-a)$ to directly apply the functional equation. By substituting $X=a$ and $Y=x-a$ and using the derived value of $f(a)$, we simplified the equation to find that $f(2a-x) = -f(x)$.

The final answer is $\boxed{-f(x)}$. This corresponds to option (A).