The question here is for what numbers y is it possible to find a number x so that y = √(x4 - 81)? The operator √ always returns a non-negative number so y must be non-negative but given a non-negative number y can you find an x so that y = √(x4 - 81)?
If you square both sides you get y2 = x4 - 81 and hence x4 = y2 + 81. The right side is positive so I can take the fourth root to get x = (y2 + 81)1/4.
You give me a non-negative number y and I take x = (y2 + 81)1/4. Then
√(x4 - 81) = √)[(y2 + 81)1/4]4 - 81) = √(y2 + 81 - 81) = √y2 and since y was non-negative √y2 = y.
Thus the range of f(x) is all non-negative numbers.