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The graph(1) $G$ of the function $f:X\to Y$ is the set $\{(x,f(x)):x \in X\}$. Unless I'm mistaken, this can be interpreted as a directed, bipartite graph(2) where every vertex in one of the partitions (domain) has out-degree $1$ and in-degree $0$. The other partition is the image.

CLARIFICATION: graph(2) refers to the graph-theoretic definition of graph, distinct from graph(1)

According to the post Difference between a function and a graph of a function?, the difference between a function $f$ and its graph $G$ is that the function's codomain cannot be recovered from the graph.

However, the domain of $f$ could be recovered from the disjoint union of $G$ with a second graph $H$ of isolated vertices which satisfies the condition -- the union of the set of vertices in $H$ and the set of vertices in the image partition of $G$ is the codomain.

Could you then say that $f$ is equivalent to $G\oplus H$ up to an isomorphism? Is there any benefit to viewing functions as a specific type of graph?

Mithrandir
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  • We usually call the second partition the “range” or “co-domain” rather than “image.” The “image” is the set of values equal to some $f(x),$ high can be a proper subset of the range. – Thomas Andrews Aug 06 '20 at 22:55
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    Also, while you can consider this to be a kind of bipartite graph, that is not the meaning of graph used in this sense. This is “graph” as in “Draw a graph of $y=x^2.$”” – Thomas Andrews Aug 06 '20 at 22:57
  • @ThomasAndrews the second partition of graph $G$ is the image, not the co-domain. This is why a function's graph alone does not tell you the co-domain. The second partition is only $Y$ if $f$ is surjective. – Mithrandir Aug 06 '20 at 23:13

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As a visualizing tool, yes, some times a graph is better than a graph. Visualizing a function $\Bbb R\to\Bbb R$ as a collection of arrows from one number line to another (or just as a device which, when you press somewhere in the domain, the corresponding place in the codomain lights up) is, in my opinion, a better framework than a curve in the plane for understanding things like differentiation, the $\varepsilon$-$\delta$ definition of continuity, and function composition (one can, for instance, give an entirely clear and obvious intuitive argument for the chain rule in a matter of seconds, once derivatives and function chaining / composition are both understood).

It is also more easily generalized and therefore often used (at least in an abstract sense) in both analysis and topology.

However, while chaining functions is basically a trivial operation, and it neatly separates domain from codomain, which is nice some times, arithmetic operations are more difficult to visualize (I don't know intuitively how to combine my images of the graph-theory graphs for $x\mapsto x^2$ and $x\mapsto x$ into their sum $x\mapsto x^2+x$, but for the curves in the plane it is obvious). Additionally, it does have the drawback of not working too well on a printed page. Which I suspect is why it is a sadly underrepresented representation in calculus classes.

Arthur
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  • I appreciated "some times a graph is better than a graph," ha. – Mithrandir Aug 06 '20 at 23:27
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    As a personal note, it was only when my analysis class visualised arbitrary functions as arrows from one "blob" to another that I actually truly grokked the $\varepsilon$-$\delta$ definition of continuity, after quite some time struggling with it in calculus. Trying to draw the definition on a standard planar graph is just messy, there are too many things going on, and I could never quite remember how all the lines were meant to interact. And trying to memorize the definition symbol by symbol is begging to mix up the order of $\forall, \exists, \varepsilon$ and $\delta$. – Arthur Aug 06 '20 at 23:29