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

1 Answer
Nov 27, 2015

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.