Why do we need this.

I am about to be teaching my students about rational functions.  The one above is one I am planning on basing a lecture around.  In all likelihood, a few students will ask why they need to know this and what it is that they need to know.  I want them to be able to explore the relationships involved in the function and how they result in the curve.  There is some application of the problem, I am not sure of what that is but one can find areas where math is used in all kinds of interesting places.  People did not arrive at these problems to solve an application, though.  The vast majority of math problems have been solved as reasons unto themselves. What purpose does this have? Math is a creation of man.  We use it to explain things around us but everything you see above is applied thought, not a reflection of observation.  Many of the courses we study are based on observation of actions and reactions of the world around us but some are extensions of personal expression.  People recognize paintings and songs as such but they like to exclude math from this group.  The do so because math takes study to understand. 
How do these forms of expression aid our knowledge? Why study them? Our minds, bodies, and souls are built from the same stuff as the rest of the universe.  Those things that we create from within are not expressions different from what we observe but internal developments in a manner that reflect the relationships we see in the world and amongst ourselves.  Math is a study of thought and provides us a possible window into the cognitive process of the everything around us.  Does this make sense? I wish I could more carefully formalize the argument and make a more profound connection.  Hopefully, through the act of teaching and learning, I can.