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 hyperbola

Graph of the inverse proportionality `y=k/x` is called hyperbola.