Question

Do the following equations define functions: (i) `y = x^2-5x` (ii) `x = y^2-5y` ?

Answer 1

See explanation...

Explanation:

`color(white)()`
First equation: `y = x^2-5x`

For the first equation, putting `x=-6` we find:

`y = x^2-5x = (-6)^2-5(-6) = 36+30 = 66`

Note that the value of `y` is uniquely determined by the value of `x`. This is true for any value of `x`, so `y` is a function of `x`.

graph{(y - x^2+5x)(x+6+0.0001y) = 0 [-11, 11, -11, 102]}

`color(white)()`
Second equation: `x = y^2-5y`

For the second equation, putting `x=-6` we find:

`-6 = x = y^2-5y`

Adding `6` to both ends we get:

`0 = y^2-5y+6 = (y-2)(y-3)`

So `y = 2` or `y = 3`

Note that `y` is not uniquely determined by the value of `x`, so is not a function of `x`.

graph{(x - y^2+5y)(x+6+0.0001y) = 0 [-12, 2, -2.5, 6]}