Official solution
\[f\left( {2a - x} \right) = f\left( {a - \left( {x - a} \right)} \right)\]
\[ = f\left( a \right)f\left( {x - a} \right) - f\left( 0 \right)f\left( x \right)\]
\[ = f\left( a \right)f\left( {x - a} \right) - f\left( x \right)\]
\[ = - f\left( x \right)\]
\[\left[ {} \right.\] as \[x = 0,y = 0,f\left( 0 \right) = {f^2}\left( 0 \right) - {f^2}\left( a \right)\]
\[\,\,\,\,\,\,\,\, \Rightarrow {f^2}\left( a \right) = 0 \Rightarrow f\left( a \right) = \left. 0 \right]\]
\[ \Rightarrow f\left( {2a - x} \right) = - f\left( x \right)\]