To talk about the inverse function, the original function f(x) = 2x^{2 }+ 8x - 7 must be injective. However, a quadratic polynomial considered as a function on all real numbers is not injective because a horizontal line can intersect the graph in two points. In particular, there are two values of x for which f(x) = 2.

The discriminant is 64 + 72.

This needs more context. Inverse of what and what is or is not an inverse? I am not sure what the connection is with testing for a perfect square. A question whether some n is a perfect square is a question about the existence of an integer (whose square is n), but the quadratic formula accepts and returns real numbers in general.

As a function from all reals to all reals, no quadratic polynomial is either injective (because the graph has two branches) or surjective (because the function has a minimum or a maximum).