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A real-valued function f(x) obeys the relation f(x-y)=f(x)f(y)-f(a-x)f(a+y), where a is a fixed constant, and f(0)=1. Then the value of f(2a-x) is

1. -f(x)
2. f(x)
3. f(a)+f(a-x)
4. f(-x)

Correct answer: -f(x)

Solution

The given functional equation can be manipulated by substituting specific values for x and y, revealing that f(2a-x) is indeed the negative of f(x). This relationship holds due to the symmetry and properties of the function as defined by the equation.

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