The reciprocal function and the hyperbola

Share a cake among guests: the more guests there are, the smaller each share gets. With 2 guests, each one gets 1/2 of the cake; with 4 guests, 1/4; with 10 guests, 1/10. This simple idea is the reciprocal function: to every nonzero number x, it assigns its reciprocal 1/x.

We write f(x) = 1/x. Watch the starting point: you cannot divide by zero, so 0 has no image. The domain of the reciprocal function is therefore all real numbers except 0, written ℝ*.

Its curve is called a hyperbola. It is made of two separate branches: one in the quarter plane where x and y are positive, the other where x and y are negative. When x gets very large, 1/x gets close to 0 without ever reaching it: the curve hugs the x-axis. When x gets close to 0, 1/x blows up: the curve hugs the y-axis. These two axes are the asymptotes of the hyperbola.

One last elegant property: the reciprocal of -x is -(1/x). The curve is therefore symmetric about the origin: we say the reciprocal function is odd. On each of its two branches it is decreasing: as x grows, 1/x shrinks.