We can prove that #f(x)# has an inverse easily enough, in that
#d/(dx)f(x) = 5x^4+1 >= 1# for all #x in RR#
so #f(x)# is strictly monotonically increasing.
The problem is that a quintic of the form #x^5+x+a = 0# does not generally have a solution expressible using normal arithmetic operations and radicals. This is where the Bring radical comes in.
The Bring radical of a number #a# is a root of #x^5+x+a = 0#.
If #a# is real, then it is the unique real root.
So if #y = f(x) = x^5 + x#, then #x^5 + x + (-y) = 0#
and #BR(-y)# is the real root of #x^5 + x + (-y) = 0#
