Definition:
We say that a function  $f$  has even  symmetry with respect to $x=h$ ("axis reflective symmetry") for a set  D, if whenever $x_a=h+a \in D$ , $x_{-a}=h-a \in D$ and  $f(h-a) = f(h+a)$.
We say that a function  $f$  has odd  symmetry with respect to the point $(h,k)$ for a set  D, if whenever $x_a=h+a \in D$ , $x_{-a}=h-a \in D$ and  $f(h-a) - k  = k - f(h+a)$.

Graph
Mapping Diagram
$f$  has even  symmetry with respect to $x=h$:
$f(h-a) = f(h+a)$
Graph:$f(h-a) =
              f(h+a)$
MD: $f(h-a) =
              f(h+a)$
$f$  has odd  symmetry with respect to the point $(h,k)$:
$f(h-a) - k  = k-f(h+a)$.
Graph:$f(h-a) - k =
              -(f(h+a) - k)$
MD:$f(h-a) - k =
              k-f(h+a)$