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

  • MHB
  • Thread starter mathdad
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In summary, an expression fails to be a function when it does not pass the vertical line test, meaning that there are multiple outputs for a single input. In other words, there are two or more values of the independent variable that map to the same value of the dependent variable. This is also known as a one-to-many relationship and is not considered a function.
  • #1
mathdad
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Why is y = x^2 not one-to-one?
 
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  • #2
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:

\(\displaystyle x=\pm\sqrt{y}\)

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.
 
  • #3
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?
 
  • #4
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
 
  • #5
I like the vertical and horizontal line tests.

Question:

When does an expression fail to be a function?
 

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

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

The graph of y = x^2 is a parabola, which opens upwards and has a vertex at (0,0). This means that for every x-value, there are two possible y-values. Therefore, the function is not one-to-one because there are multiple y-values for some x-values.

How can I tell if a function is one-to-one?

A function is one-to-one if every x-value has a unique y-value. This means that no two different x-values can have the same y-value. To determine if a function is one-to-one, you can graph it and see if it passes the horizontal line test. If a horizontal line intersects the graph at more than one point, then the function is not one-to-one.

Can y = x^2 ever be one-to-one?

No, the function y = x^2 can never be one-to-one because of its shape. A parabola will always have two y-values for each x-value, unless it is a straight line (which is not the case for y = x^2).

Why is it important for a function to be one-to-one?

One-to-one functions have a unique inverse, meaning that the input and output values can be switched to create another function. This can be useful for solving equations or finding the original value of a function. Additionally, one-to-one functions have a clear and predictable relationship between the input and output values, making them easier to work with and analyze.

Can a function be one-to-one but not onto?

Yes, a function can be one-to-one but not onto. Being one-to-one means that each input has a unique output, but it does not guarantee that every output has a corresponding input. Being onto, or surjective, means that every output has a corresponding input. A function can be one-to-one but not onto if there are some outputs that do not have an input that maps to them.

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