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Is there a hard and fast way, step by step process to figure out domain and range? I don't know where to start and lack the insight to just know what it can and can't be.

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Is there any certain function in your mind? –  B. S. Aug 4 '13 at 11:57
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The domain $A$ and the codomain $B$ are inherent parts of a function $f: A \to B$, so they can just be read off the definition of $f$.

Maybe your question is about how to extract domain and range from the graph of $f : A\to B$, which is the subset $G = \{(a,f(a)) \mid a\in A\}$ of $A\times B$. The domain $A$ can be recovered from the graph as $$\{a \mid \exists b : (a,b)\in G\}.$$ For the range, it depends what you are referring to: There is no way to recover the codomain $B$ from the graph. The image of the function is given by $$\operatorname{im}(f) = \{f(a) \mid a\in A\} = \{b \mid \exists a : (a,b)\in G\}.$$

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