Because therearepoints where you cannot define a tangent vector to the given curve (there are "corners" at, (0, 0), (1/4, 4), and (1, 1)), one formula will not work. Because smooth functions are so nice, our ways of writing formulas has developed to give smooth functions- and this boundary cannot be described "smoothly".

Yes, you will need to use a "piecewise" function, with separate pieces for y= 4x, y= x, and y= 1/x. You will have to be careful to make the "pieces" join at the same values of the parameter.