.
Look into the plot of the function y = x^(x+1).
The function is between 0 and 1 in the interval 0 < x <= 1.
It is the domain of your function.
Plot y = x^(x+1)
It is informal demonstration, but the formal proof is easy, too.
(1) The function y = x^(x+1) is determined for non-negative values of x, ONLY.
(2) If 0 < x <= 1 then x^(x+1) is OBVIOUSLY less or equal to 1.
(3) If x > 1 , then x^(x+1) is OBVIOUSLY greater than 1, and the expression under the square root is negative.
These three positions complete the proof.
