# Inverse Proportionality

It is said that y is inversely proportional to x if their product is constant, i.e. xy=k or y=k/x .

Inverse proportionality is a function of the form y=k/x, where k!=0. Number k is called coefficient of inverse proportionality.

Properties of the function y=k/x:

1. Domain of the function is all number line (set R of real numbers) except x=0.
2. Function is odd, because f(-x)=k/(-x)=-k/x=-f(x).
3. When k>0, function is decreasing on (-oo,0) and (0,+oo), when k<0, function is increasing on (-oo,0) and (0,+oo).
4. x-axis and y-axis, i.e. lines x=0 and y-0, are vertical and horizontal asymptotes of the graph. This means that graph approaches (asymptotically) coordinate axis.

Let's draw graph of the function y=1/x. For this first draw a graph on the interval (0,+oo): let's choose a couple of points and find value of function at these points.

 x 1/4 1/2 1 2 4 mathbf (f(x)=1/x) 4 2 1 1/2 1/4

Now draw these points on coordinate plane and connect them with smooth curve. This is part of y=1/x on interval (0,+oo). Now, using the fact that function y=1/x is odd, we draw another part symmetric about the origin to the drawn one. So, we've obtained graph of the function y=1/x.

Similar form have graphs of the function y=k/x when k>0.

If k<0 then parts of graph should be drawn not in I and III quadrants, but in II and IV (see figure below). Graph of the inverse proportionality y=k/x is called hyperbola.