As shown in Figure 1, it is known that the images of positive scale function and inverse scale function all pass through the point m (- 2, - 1), and P (- 1. - 2) is a point on the hyperbola, and Q is the coordinate plane

As shown in Figure 1, it is known that the images of positive scale function and inverse scale function all pass through the point m (- 2, - 1), and P (- 1. - 2) is a point on the hyperbola, and Q is the coordinate plane

(1) Let the analytic expression of positive scale function be y = KX, and the analytic expression of inverse scale function be y = m / X
-2k=-1,k=1/2,m=2,
Therefore, the analytic expressions of the two functions are: y = x / 2, y = 2 / X
(2) Existence
When q is on the straight line OM, its coordinates can be set as (x, X / 2), s △ AOP = AP * OA / 2 = 2 * 1 / 2 = 1
S △ OBQ = s △ AOP, that is | x | * | X / 2 | / 2 = 1, x = 2 or - 2,
The coordinates of Q point are (2,1) or (- 2, - 1)
(3) The coordinate of point q is (x, 2 / x) on the part of hyperbola y = 2 / X in the first quadrant. The perimeter of parallelogram opcq should be the smallest. Because the length of OP is fixed, it is root 5, so as long as the length of OQ is the smallest
OQ square = x ^ 2 + (2 / x) ^ 2 = (X-2 / x) ^ 2 + 4. When x = 2 / x, the square of OQ has the minimum value, then x = root 2 (negative value is rounded off),
The minimum value of OQ is 2, so the minimum value of perimeter is 4 + 2 root sign 5