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Initial question :

sage: with assuming(a,"noninteger"):(xsn==prod).solve(p_x)
[p_x == c^(1/(a + 1))]

Second question :

sage: with assuming(x>0, a,"noninteger"):
....:     fp_x(x)=(x==(1/p_x)^a).solve(p_x)[0].rhs()
sage: fp_x
x |--> 1/(x^(1/a))

A generic solution to this second question raises a lot of questions :

  • Is f an expression or a callable expression ?

  • What is the relevant argument ?

  • What are the necessary assumptions ?

And there are probably others...

Generic notes :

  • You should lighten and densify your notations (e. g. var("x, y") ; assume (x, "real") ; assume(x>0) ; assume(y, "real") ; assume (y>0) can be lightened into var("x,y", domain="positive").

  • You should distinguish what you want to use as expressions and what you deem functions, distinguishing further arguments and parameters in the latter case...

Learn Python ; learn Sage (a sound time investment, in both cases...). But I repeat myself...