Question

does x squared -y=9 define a function of x?

Answer 1

To be a function of x, there must be only one value of y for any value of x.

\[x^2-y = 9\]\[y=x^2-9\]For any value of x, there's only one value of y.

If instead we had an equation like\[y^2 - x = 9\]\[x = y^2-9\] we would not have a function. Consider \[x = 0 = y^2-9\]\[y=\sqrt{x+9} = \sqrt{0+9}\ = \pm 3\]Both y = 3 and y = -3 satisfy the equation at x = 0. Therefore, \[x = y^2 -9\] is not a function.