Using your relation: $(x, y) \in S \subset\mathbb{R^2}, \;\;xSy \iff y=\left|\left(\frac13\right)^x-1\right|,\quad x, y \in \mathbb{R}$ you need to determine if the following properties hold:
Reflexivity:
Is $x S x$ for all $x \in \mathbb{R}$? If so, reflexivity holds. If not, then reflexivity fails.
What can you say about whether or not $x = \left|\left(\frac13\right)^x-1\right|$ is true for all $x$ in $\mathbb{R}$?
Symmetry:
if $x S y$, is $y S x$ for all $x, y \in \mathbb{R}$?
If so, symmetry holds. If not, it fails.
Is it always the case that for $(x, y) \in \mathbb{R}^2$, if $y = \left|\left(\frac13\right)^x-1\right|$, then $x = \left|\left(\frac13\right)^y-1\right|$?
If not, then the relation $S$ is not symmetric.
Antisymmetry:
for all $x, y, z \in \mathbb{R}$, if $x S y$ and $y S x$, does this imply that $x = y$?
If so, the relation is antisymmetric. If not, the relation is not antisymmetric.
Is it always the case that if $y = \left|\left(\frac13\right)^x-1\right|$ and $x = \left|\left(\frac13\right)^y-1\right|$, then it follows that $x = y$?
Transitivity:
If $x S y$ and $ySz$, is $x S z$, for all $x, y, z \in \mathbb{R}$?
If so, then the relation is transitive; if not, then the relation is not transitive.
Is it always the case that if $y = \left|\left(\frac13\right)^x-1\right|$, and $\left|\left(\frac13\right)^y-1\right|$, then it follows that $z = \left|\left(\frac13\right)^x-1\right|$?
Unpack, in a similar manner, exactly what is required for a relation to satisfy the properties of completeness and asymmetry (i.e., use the definitions of a completeness and asymmetry, and test whether your relation $S$ meets the required conditions for those properties to hold.)
Note: if you can find any counterexamples to any given property, you thereby show that the a property doesn't hold for $S$, because a relation only has a property if it is holds for all elements of the set on which it is defined. Put differently, a property hold unless there is a counterexample that satisfies the "if(s)"..., but fails to satisfy the "then" part of the property's definition.