Why is y = x^2 not one-to-one?

[mathdad]

Why is y = x^2 not one-to-one?

 

[MarkFL]

Consider y = 1...there are two inputs (x = -1 and x = 1) that map to y = 1. And so y = x² is not one-to-one. :D

If we try to solve for x, we get:

$$x=\pm\sqrt{y}$$

This tells us that for a particular y greater than zero, we have 2 x's that map to it...unless we restrict x (the domain) such that it is either non-negative or non-positive.

 

[mathdad]

Ok. If we let x = 1 or -1 for y = x^2, both values lead to y = 1 after squaring. We can also say that y goes to 1 for both values of x. The conclusion is that the parabola y = x^2 is a function but not one-to-one.

Correct?

 

[MarkFL]

The conclusion is that the parabola y = x^2 is a function but not one-to-one.

Correct?

It passes the vertical line test, and so is a function, but fails the horizontal line test, and so is not one-to-one. :D

 

[mathdad]

I like the vertical and horizontal line tests.

Question:

When does an expression fail to be a function?