Symmetry in functions What's the difference between $f(x)=f(a-x)$ and $f(x)=f(x-a)$ ?
It's a pretty simple question maybe, but I'm unable to understand this one. 
 A: They mean two different things, and without knowing what it is that you don't understand, it's hard to know how to explain.
The way to understand it is to abandon algebra. Put $a=5$ and try different values of $x$: 0, 1, 2, 3, 4, 5 and so on.
You will find that:


*

*$f(x)=f(a-x)$ means what it says. For example, that $f(0)=f(a)$ and $f(-1)=f(a+1)$. To summarize: $f$ is mirror-symmetrical about the value $x=\frac12{a}$.

*$f(x)=f(x-a)$ means what it says. For example, that $f(a)=f(0)=f(-a)$. To summarise: $f$ is periodic with period $a$.
Note for pedants: $f(x)=f(x-a)$ actually means that $f$ is periodic with period $|a|$ (that is, $a$ if $a$ is positive, $-a$ if $a$ is negative), and if $a=0$ if means $f(x)=f(x)$, which is always true but very uninteresting. 
A: The sign "$-$" here refers to a binary operation, lets call it $O(x,y)=x-y$. Writing it like this shows the difference between $O(x,y)$ and $O(y,x)$
So the difference is $O(x,y)\neq O(y,x)$ since a binary operation does not have to be commutative, the result could be different. 
There are a lot of examples of non commutative operation:


*

*Composition of functions $f\circ g\neq g\circ f$.


2.Division $4/2\neq 2/4$.


*Multiplication of matrices (with at least 2 rows).


To read more about non commutative binary operations I would suggest you read about basic Relations in Set Theory and about basic Group Theory.
