Find an elementary function of calculus that maps an interval one to one onto .
We can build that function geometrically and represent intuitive look of such function.
is the axis. We curve to half o a circle which his diameter parallel to .
We see every point as a point on half circle.
Now let us define :
For every we will pass straight line joining the center of the circle with point (which is on the half circle) and continue to axis(which is the whole .
The intersection point which that line makes with axis is .
It easy to see that f is one-to-one from onto .