1.2 Exercise 6

This was a really easy problem, that ends the section on functions in Munkres's text.  The next fifteen problems deal with relations, and I am finding them immensely interesting.

"Let  f : \mathbb R \rightarrow \mathbb R be the function  f(x) = x^3 - x .  By restricting the domain and range of  f appropriately, obtain from  f a bijective function  g .  Draw the graphs of  g and  g^{-1} . (There are several possible choices for  g .) "

(Taken from Topology by James R. Munkres, Second Edition, Prentice Hall, NJ, 2000. Page 21.)

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SOLUTION  

As an example,  g: (-\infty, -1) \rightarrow \mathbb{R}^{-} with rule  g(x) = x^3 -x is bijective.

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