Is |x^3-1| One to One? Monotonicity and Inverse Function Analysis

[peripatein]

Homework Statement

Is the function |x^3-1| one to one? Is it monotonous?

Homework Equations

The Attempt at a Solution

Since |x^3-1|=|y^3-1| does not necessarily imply that x=y for every x and y, I presume it is not one to one. Hence it has no inverse function.
It is also not monotonous.
Are all these statements correct?

 

[tiny-tim]

yes

 

[peripatein]

Hi tiny-tim,
Once again, thanks a lot! :-)

 

[micromass]

While the statements are correct, they are not sufficient to complete the exercise. To actually solve it, you need to find counterexamples. For example, if you want to show that $|x^3-1|$ is not one-to-one, you need to come up with two particular and distinct points x and y such that $|x^3-1|=|y^3-1|$. Just saying that it is one-to-one is not enough without counterexample.

 

[peripatein]

Have done so, simply didn't specify it :-). Thank you, micromass!