When we consider y^2 = x, we typically rewrite it as y = ±√x. I know the two meanings are close enough to work in most cases. But is there a technical distinction between them? In its purest definition, what does y^2 = x mean?
by 
Doug138
ANSWERS: 2
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y^2=x is a parabola (like y=x^2) except instead of going the direction of the positive y axis, it goes along the positive x axis. The problem comes in that if the problem is kept in the form y^2=x then it is impossible to come up with negative results for x when you plug in any number for y. Thus you end up with half of the sideways parabola.
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If x and y are variables, then y^2 = x expresses a relationship between them. If one variable is a function of the other then additional criteria apply. y = √x is a function in the form y = f(x) because the square root is defined to be non-negative, yielding a unique value. The relation y = ± √x gives all values satisfying y^2 = x, but fails as a function because it's multi-valued. If x and y are Cartesian coordinates, then the graphical distinction is between a complete parabola or only one of its limbs.
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