x-1=y^2 Does the equation define y as a function of x?

No.

I agree with the other answers that have said it expresses x as a function of y...but in principle, an equation could both define x as a function of y and define y as a function of x.

The reason this equation does not "express y as a function of x" is that there are two possible y-values corresponding to the same choice of x. The points (2,1) and (2,-1) both satisfy the given equation, so the input "2" does not point to a unique output. So, the solution set of this equation violates one of the defining properties of "function", often called the vertical line test.

If you were to restrict the range of the function (to, say, non-negative numbers, but there are other ways to do it), then I would then say Yes.

One would normally think of the equation as defining x as a function of y, x = y^2 - 1.

However, you can turn it around to do the opposite

x - 1 = y^2

y = ± √(x - 1) = ± (x - 1)^1/2 where y is a function of x. Notice that this form tells you the function is symmetrical, not surprising because it can be expressed as a quadratic.

it is just a matter of which is the independent variable

x - 1 = y²

x = y² + 1

The function defines x as a function of y and graphs as a right-opening parabola with vertex (1, 0).