When solving square roots, don't we have to consider the positive or negative solution?
Yes, if you solve y^2 = 2 - x for y, by taking the square root of each side, then you do need to consider both the square root of 2-x and its opposite. This generates two functions, as explained below.
In the relationship between x and y given by the equation
x + y^2 = 2
it's true that y is not a function of x. Rather, as JeffM posted, x is a function of y.
However, if we restrict the values of y to be non-negative, then y is a function of x.
Alternatively, if we restrict y to be negative, then y becomes a different function of x.
y1(x) = sqrt(2 - x)
y2(x) = - sqrt(2 - x)
The domain for each of these functions is all numbers less than or equal to 2.
The range for y1(x) is all non-negative numbers.
The range for y2(x) is all negative numbers and zero.
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