I am familiar with the contraposition definition of if P then Q and if Q then P, as well as real world examples such as "if all cats are animals" then "if something is a cat, it is an animal" . However I am confused when applying it t to something more abstract such as:
“If x^2 is even, then so is x”, where x is an integer
where do I begin to find the contraposition of the statement?
The contrapositive of "if $P$ then $Q$" is "if not $Q$ then not $P$".
What you have is "if $x^2$ is even, then $x$ is even", so with $P$ as "$x^2$ is even" and $Q$ as "$x$ is even", the contrapositive is "if $x$ is odd, $x^2$ is odd".