Don't understand results from solving polynomial equation

Question

I attempting to solve an equation for x so that I may then switch x & y to get the inverse function. This is the code I have for solving for x:

f[x_] := -3 + (x^3)/1367 + (x^5)/2734        
mySolution = Solve[f[x] == y, x]

When I execute this code, Mathematica outputs:

{{x -> Root[-8202 - 2734 y + 2 #1^3 + #1^5 &, 1]}, 
 {x -> Root[-8202 - 2734 y + 2 #1^3 + #1^5 &, 2]}, 
 {x -> Root[-8202 - 2734 y + 2 #1^3 + #1^5 &, 3]}, 
 {x -> Root[-8202 - 2734 y + 2 #1^3 + #1^5 &, 4]}, 
 {x -> Root[-8202 - 2734 y + 2 #1^3 + #1^5 &, 5]}}

Could someone help me understand why this is the output when solving the function for x? I don't see how I could switch x & y with the output Mathematica has given me.

What do you expect? There will be multiple values for x in f that have the same result, and these are the roots indicated. — ciao, Sep 16 '15 at 23:15
Your equation is equivalent to solving fifth order polynomial and in general it is impossible to solve such equations in terms of radicals. For more on these toppics read e.g. How do I work with Root objects? and What is the difference between Reduce and Solve?. There are many more related posts, nonetheless your question is a duplicate of the first link. — Artes, Sep 16 '15 at 23:17
By the way objects like -8202 - 2734 y + 2 #1^3 + #1^5 & represents so called pure functions and you can find posts discussing related aspects. — Artes, Sep 16 '15 at 23:22
@artes With the comment below the answer in your link it looks like there's a problem in v10. Did you check that? — Sjoerd C. de Vries, Sep 17 '15 at 05:38
@SjoerdC.deVries What kind of problems have you found? E.g. Solve[(3 - Cos[4x])(Sin[x] - Cos[x]) == 2, x, InverseFunctions -> True] yields slightly different form of solutions then in previous versions, nevertheless both are equivalent. — Artes, Sep 17 '15 at 07:38
@artes I haven't done anything here. I was referring to a comment below the answer of yours you linked to. That seems to suggest a problem in v10 and it looked like you hadn't seen the comment. — Sjoerd C. de Vries, Sep 17 '15 at 11:30

Don't understand results from solving polynomial equation

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