Question

How do you decide whether the relation `x = y^2 - 2y + 1` defines a function?

Answer 1

This relation is not a function.

Explanation:

Your relation is not a function.

The reason for this is that you can find an `x` where `y` doesn't have a unique value.

Let's transform the right side a bit so that it's easier to see:

`x = (y-1)^2`

Now, if this relationship was a function, you would need to obtain one unique ("one and only one") value for `y` for any value of `x in RR`.

However, this doesn't hold here:

E.g., for `x = 5`, both `y = 6` and `y = -4` provide the equality:

`5 = (6-1)^2` and `5 = (-4-1)^2` are both true.

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Another way to see this is to graph the function:

graph{x = y^2 - 2y + 1 [-5, 15, -5, 5]}

Here, you also see very clearly that:

• the function is not defined for `x <0`,
• for `x = 0` it has indeed one unique value for `y`,
• and for `x > 0`, each `x` has two `y` values.