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I have an exam tomorrow in Discrete Math. One of the questions on the review sheet was this: 

Describe a one-to-one and onto function from the real numbers in (-1,-1) to ℝ?

I am currently try to think of a function that satisfies this condition, however, I am having difficulty. Would someone mind providing me with a function that satisfies these conditions?

Thanks

Grant
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  • $(-1,-1)$ or $(-1,1)$? –  May 08 '14 at 02:25
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    Just do something with the tangent or cotangent function. –  May 08 '14 at 02:28
  • It sounds like the question isn't even asking for a function; just a description of what it would look like. If you restrict yourself to continuous functions, simply knowing the possible descriptions of such functions is extremely informative. –  May 08 '14 at 02:54

1 Answers1

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For mapping $(-1,1)$ bijectively to $\mathbb{R}$, the standard example is $\tan\left(\frac{\pi x}{2}\right)$.

Another function that works nicely is $\dfrac{x}{1-|x|}$. Or, if you prefer, $\dfrac{x}{1-x^2}$.

André Nicolas
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    @MattAllegro: As $x$ climbs from $0$ to $1$, $\tan(\pi x/2)$ climbs from $0$ to "$\infty$." You are probably thinking of the inverse function. – André Nicolas May 08 '14 at 02:51