Well, whoever told you that was wrong!
I suspect they graphed that on a calculator and saw no graph for x< 1.
Many perfectly good calculators (like my TI 89) will not handle odd roots of negative numbers correctly because they do them using a logarithm which is not defined for .
If you graph y1= (x-1)^(4/5)*x on a TI calculator, for example, you will see a graph that begins at (1, 0) and rises to the right.
If you also graph y2= (1- x)^(4/5)*x, you will see the rest of the graph which has maximum at x= 5/9.