First, the inverse function is y = ((x - 1)^2 - 2) / 3. Second, how can the domain of the inverse function be all reals if the original function returns only values >= 1? Formally, when you are solving the equation to find the inverse function, note that x - 1 = sqrt(2 + 3x) is not equivalent to (x - 1)^2 = 2 + 3x. More precisely, every pair (x, y) that satisfies the first equation also satisfies the second one but not vice versa. This is why you end up with an equation that has more solutions (x, y) than the original one.