Question

Does the equation x^2+y^2=100 define y as a function of x

Answer 1

`x^(2) + y^(2) = 100` does not define
`y` as a function of
`x`.

Explanation:

In a function, for every valid input value
`x`, there should only be one unique output value
`y`.

For example, consider a mapping that includes both
`(6,\, 8)` and
`(6,\, -8)`. This mapping isn't a function. The reason is that the input
`x = 6` is mapped to more than one output values:
`y = 8` and
`y = (-8)`.

For the given equation
`x^(2) + y^(2) = 100`, there might be more than one possible output values
`y` for some input values
`x`. The reason is that
`y^(2) = (-y)^(2)`.

For example, if the input value is
`x = 6`, then both
`y = 8` and
`y = (-8)` (output values) would satisfy this equation:

`6^(2) + 8^(2) = 100`.

`6^(2) + (-8)^(2) = 100`.

Hence, the mapping defined with
`x^(2) + y^(2) = 100` (with
`x` as input and
`y` as output) is not a function.