Given the graph of $y = \frac{1}{x}$, construct the $(x,y)$ coordinate axes using a straight edge and a compass.
The solution to this problem is known (mouse over the spoiler text below for a hint).
Straight edge and compass as in classic Euclidian geometry. There is no way to measure distance. The $y = \frac{1}{x}$ function has an interesting property: if you choose one part of the graph (say $x>0$) and have two distinct parallel segments on it, the line that goes through the midpoints of those segments has to pass through the origin. After you’re able to show that, it’s not hard to figure out the rest.
What other analytic functions can one substitute for $y = \frac{1}{x}$, and still be able to do so?