Question

{"given":"A real valued function $f(x)$ satisfies the functional equation $f(x - y) = f(x)f(y) - f(a - x)f(a + y)$ where $a$ is a given constant and $f(0) = 1$. We need to find $f(2a - x)$.","key_observation":"The key is to use strategic substitutions in the functional equation to determine $f(a)$ first, then use the properties of the equation to establish a relationship for $f(2a - x)$. By setting specific values for $x$ and $y$, we can derive that $f(a) = 0$ and then use this to find the desired expression.","option_analysis":[{"label":"(A)","text":"$-f(x)$","verdict":"correct","explanation":"By setting $x = 0, y = 0$ in the functional equation: $f(0) = f(0)^2 - f(a)^2 = 1 - f(a)^2$. Since $f(0) = 1$, we get $f(a) = 0$. Using further substitutions and the symmetry properties of the functional equation, we can show that $f(2a - x) = -f(x)$."},{"label":"(B)","text":"$f(x)$","verdict":"incorrect","explanation":"If $f(2a - x) = f(x)$, then the function would be symmetric about $x = a$. However, the functional equation structure and the constraint $f(a) = 0$ lead to an antisymmetric relationship instead."},{"label":"(C)","text":"$f(a) + f(a - x)$","verdict":"incorrect","explanation":"Since we can prove that $f(a) = 0$, this option would reduce to $f(a - x)$. However, the functional equation leads to $f(2a - x) = -f(x)$, not $f(a - x)$."},{"label":"(D)","text":"$f(-x)$","verdict":"incorrect","explanation":"This would suggest that $f(2a - x) = f(-x)$, which would mean the function has a specific symmetry property. However, the functional equation actually yields $f(2a - x) = -f(x)$, not $f(-x)$."}],"answer":"(A)","formula_steps":[]}