UNO High School Problem of the Week Competition
Problem 4
A point x is called a fixed-point of a function f if f(x) = x. For example, the function f(x) = x
has infinitely many fixed-points and the function f(x) = x2 has two fixed-points.
The concept can be extended to functions of more than one variable, if f(x,y) = (x,y). Here the function
maps points on the plane to points on the plane.
Find the fixed-points of the function f(x,y) = (x3 + 2x2 - 2, y2 - xy + x ).
Solution
For a point (x,y) to be a fixed-point of f, it must be the case that
x = x3 + 2x2 - 2
y = y2 - xy + x
Now the x-coordinate equation can be rewritten and factored into (x2-1)(x+2).
Therefore the possible values of x are -2, -1, and 1.
When x = -2, the y-coordinate equation is y2 + 2y - 2 = y, which when rewritten and factored, is
(y+2)(y-1) which has solutions of y = -2 and y = 1. So two of the fixed-points are (-2,-2) and (-2,1).
When x = -1, the process is similar to when x = -2, and leads to the the equation y2 - 1 = 0,
or that y = -1 and 1. So two other fixed-points are (-1,-1) and (-1,1).
Finally when x = 1, the y-coordinate equation becomes y2 - 2y + 1 = 0, or when factored, is (y-1)2=0,
meaning that y has a double root of y = 1. So the final fixed-point is (1,1).
Therefore f has 5 fixed-points, (-2,-2), (-2,1), (-1,-1), (-1,1), and (1,1).