If $R$ is transitive, then $R^+$ exists. In fact, $R^+ = R$.
So for this question. I was wondering If I had the correct approach. My thinking went like this: Suppose $R$ is transitive.
$R^+$ means it is the smallest transitive x that contains $R$ as a subset. By definition, $R^+$ is the smallest transitive of $R$. It contains itself as a subset. So therefore, $R^+ = R$.
The idea you present is in spirit correct, but you need to write more precisely. For instance you write
but probably mean "$R^{+}$ is the smallest transitive relation on the set $X$ that contains $R$ as a subset".
means exactly the same thing as that which you wrote in the previous sentence. And by continuing with
You are only saying that "$R^{+}$ contains itself as a subset" which is probably not what you wanted to say but this in turn does not justify the punchline $R^{+}=R$.
As such it is difficult to judge the correctness of what you have written.
If I were to write up the idea which I think you are trying to convey it would be:
Suppose that $R$ is a transitive relation on a set $X$. By definition, $R^{+}$ is the smallest transitive relation on $X$ that contains $R$ as a subset. Since $R$ is a transitive relation on $X$ that contains $R$, $R^{+}\subseteq R$. Then we get $R\subseteq R^{+}\subseteq R$ so $R^{+} = R$.