



 
Hi Cane, I'm going to illustrate with the function
For a number x to be in the domain of y you need both x^{2}  x ≥ 0 and x  x^{2} ≥ 0. Think of the graph of y = x^{2}  x. If y = x^{2}  x changes from being above the xaxis (the function being positive) to being below the xaxis (the function being negative) the graph must pass through zero. Thus the function can only change sign at a point where x^{2}  x = 0, that is x = 0 or x = 1. This divides the xaxis into 3 segments, x < 0, 0 < x < 1, 1 < x . The expression (x^{2}  x) can't change sign in these segments since it can only change sign at x = 0 or x = 1. Now try a point in each segment.
Hence x^{2}  x ≥ 0 if x ≤ 0 or if x ≥ 1. A similar argument with x  x^{2} shows that x  x^{2} ≥ 0 if 0 ≤ x ≤ 1. Thus to ensure both x^{2}  x ≥ 0 and x  x^{2} ≥ 0 x must satisfy x ≤ 0 or if x ≥ 1, and 0 ≤ x ≤ 1. The only values of x that qualify are x = 0 and x = 1. Thus the domain of y = √(x^{2}  x)  √(x  x^{2}) consists of two points x = 0 and x = 1. When x = 0, y = √(x^{2}  x)  √(x  x^{2}) = 0 and when x = 1, y = √(x^{2}  x)  √(x  x^{2}) = 0. Thus the range of y = √(x^{2}  x)  √(x  x^{2}) consists of one point, y = 0. I hope this helps,  


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