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A warning, I've labeled this a soft question, because I'm not quite sure I know what I'm asking.

In general, the study of some mathematical structure X (a function, a shape, whatever) becomes more interesting when you introduce functions on X. It seems these functions allow ones imagination to carry over between structures.

Functions can also provide a context for what is even well defined. A triangle in a plane's angles are preserved under translations and rotations, but not continuous deformation. I've heard it said that geometry is the study of those things which don't change under translation and rotation.

More recently I was introduced to evaluation in computer interpreters, as a Monoid Homomorphism from trees. I was already vaguely familiar with differential equations being a map between functions. This idea of mapping, seems to be an incredibly powerful tool for understanding.

I would like to understand why functions are so important, and maybe improve my technical ability with them. Is there a subject for this? Or some nice reading? Or is this question too general?

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    Perhaps linear algebra and linear transformations would be a good place to get started, but I send – 2017-01-29
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    Maybe linear algebra and linear transformations would be a good place to start, but I sense your level might be more advanced that that. – 2017-01-29
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    I've been reviewing Algebra with Lang to see if I missed something. There are so many important Algebraic "blah" subjects. This has definitely helped, but I was hoping someone more experienced might be able to give more context and direction. – 2017-01-30
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    Look at category theory, maybe? – 2017-01-31
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    @AkivaWeinberger Mac Lane is on my todo list. And Category theory did partially inspire this question. Before going down that rabbit hole too far, I wanted a wider spectrum of perspectives. – 2017-01-31

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In Benedict Gross's online Abstract Algebra lecture series video 4, linked here at 12:49 and here. He claims homomorphisms are morphisms in the category of groups. Explains if he was doing everything in category theory, the objects would play almost no role. And that he would therefore encapsulate the properties into homomorphisms. Because category theory is very abstract, one will see these properties across all of mathematics. Ergo, subjects get more interesting when you introduce maps between structures.

Akiva Weinberger's suggestion was on point. The above seems to directly address my question.