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`:
- Domain of the function is all number line (set R of real numbers) except x=0.
- Function is odd, because `f(-x)=k/(-x)=-k/x=-f(x)`.
- When k>0, function is decreasing on `(-oo,0)` and `(0,+oo)`, when k<0, function is increasing on `(-oo,0)` and `(0,+oo)`.
- 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.