MHB Why is y = x^2 not one-to-one?

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The function y = x² is not one-to-one because multiple inputs yield the same output; for example, both x = -1 and x = 1 result in y = 1. This can be demonstrated mathematically, as solving for x gives x = ±√y, indicating two x-values for any positive y. While y = x² passes the vertical line test, confirming it is a function, it fails the horizontal line test, which confirms it is not one-to-one. The discussion highlights the importance of domain restrictions to achieve a one-to-one function. Overall, y = x² is a valid function but does not meet the criteria for being one-to-one.
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Why is y = x^2 not one-to-one?
 
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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.
 
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?
 
RTCNTC said:
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
 
I like the vertical and horizontal line tests.

Question:

When does an expression fail to be a function?
 

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