# Thread: Find an elementary function

1. ## Find an elementary function

Find an elementary function of calculus that maps an interval one to one onto $\displaystyle \mathbb{R}$.

2. ## Re: Find an elementary function

By "interval", you mean a bounded interval? For example, $\displaystyle \tan :\left]-\frac{\pi}2,\frac{\pi}2\right[\rightarrow \mathbb R$.

3. ## Re: Find an elementary function

Originally Posted by alexmahone
Find an elementary function of calculus that maps an interval one to one onto $\displaystyle \mathbb{R}$.
$\displaystyle f:{(0,1)} \to \mathbb{R}$

$\displaystyle f(x)=\frac{1}{x}+\frac{1}{x-1}$

4. ## Re: Find an elementary function

Originally Posted by alexmahone
Find an elementary function of calculus that maps an interval one to one onto $\displaystyle \mathbb{R}$.
We can build that function geometrically and represent intuitive look of such function.

$\displaystyle \mathbb{R}$ is the $\displaystyle x$ axis. We curve $\displaystyle (0,1)$ to half o a circle which his diameter parallel to $\displaystyle x$.

We see every point $\displaystyle x\in (0,1)$ as a point on half circle.

Now let us define $\displaystyle f(x)$:

For every $\displaystyle x\in (0,1)$ we will pass straight line joining the center of the circle with point $\displaystyle x\in (0,1)$ (which is on the half circle) and continue to $\displaystyle x$ axis(which is the whole $\displaystyle \mathbb{R}$.

The intersection point which that line makes with $\displaystyle x$ axis is $\displaystyle f(x)$.

It easy to see that f is one-to-one from $\displaystyle (0,1)$ onto $\displaystyle \mathbb{R}$.