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

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Discussion Overview

The discussion revolves around the question of why the function y = x² is not considered one-to-one. Participants explore the implications of the function's behavior, particularly in relation to its graphical representation and the concept of mapping inputs to outputs.

Discussion Character

  • Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants note that for y = 1, there are two inputs (x = -1 and x = 1) that map to the same output, indicating that y = x² is not one-to-one.
  • Others highlight that solving for x gives x = ±√y, which shows that for any y greater than zero, there are two corresponding x values unless the domain is restricted.
  • It is mentioned that while the parabola y = x² passes the vertical line test (indicating it is a function), it fails the horizontal line test, which is used to determine if a function is one-to-one.
  • A participant expresses appreciation for the vertical and horizontal line tests and raises a question about when an expression fails to be a function.

Areas of Agreement / Disagreement

Participants generally agree that y = x² is a function but not one-to-one. However, there is no consensus on the broader implications of when an expression fails to be a function, as this topic is introduced as a question rather than a settled point.

Contextual Notes

The discussion does not address specific definitions of functions or the conditions under which an expression may fail to be classified as a function, leaving these aspects unresolved.

mathdad
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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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