While studying a chapter called price elasticity of demand in my economics course, I have been presented with something called a unit elasticity curve (some sort of a hyperbola), which has supposed applications to the subject.
So being curious, I tried to find an explicit function that would fit the described properties of this curve (the book doesn't give an explicit formula and my teacher doesn't know).
The critical property of the hyperbola should be that when given a point (x,f(x)) (which is on the line), a, lets say, 10% increase in the magnitude of x, results in a 10% decrease in the magnitude of f(x), and vice versa (hence the quotient %x/%y = -1).
Clearly, if I were to plot the points resulting from the 10% increases and decreases, I would get a function (with asymptotes), but if i were to start plotting with 5% increases and decreases I would get a slightly different (though also asymptotic) function. But my book suggests that there should only be one function! God, if I were to "plot with >100% changes, it wouldn't even be a hyperbola (asymptotic) at all!
So to solve this, I treated the % change as an infinitesimal, which gives me: (dx/x)/(dy/y) = -1, next: dy/dx = -y/x. So, I guessed xy = c as the function, but this does not satisfy the previously stated property of %x/%y = -1 for non infinitesimal percentile changes.
What am I doing wrong and what is the actual explicit function? (forgive me if I am being very confusing; if there are any extra details you would like to know, just comment) Thank you very much!