Function Diagram for y=1/x

The input-output lines for the function y=1/x are all tangent to an ellipse, as you can see in this animation:

• To stop/start the animation: click once on the figure.
• You can move the x with the mouse, and see what happens to y.
• You can change the scale by moving the 1.

In the particular case where the distance between the axes equals double the unit, the ellipse is a unit circle, and the result can be deduced from theorems in Euclidean geometry. See if you can work it out, using the figure:

• You can move the x with the mouse, and see what happens to y.

Outline of a proof: use the Pythagorean Theorem to show that the length of the input-output line is x+1/x; use the Pythagorean theorem to find the lengths Qx and Qy, and its converse to show that the input-output line is the hypotenuse of a right triangle whose right angle is at Q; drop the altitude from Q to the input-output line, and show that its length is 1; conclude that the input-output line is perpendicular to a radius of the circle, and thus is tangent to it.

The result can then be generalized to any distance between the axes through a vertical or horizontal stretch, which preserves tangency. (I was told that this is a particular case of a much more general result from projective geometry: the envelope of the input-output lines is a conic section whenever the function preserves the cross-ratio. I've certainly seen many function diagrams with what seems to be a hyperbola for an envelope.)