The function f(x,y) accepts an ordered pair as input and gives another ordered pair as output. It is defined according to the following rules: If x>4, f(x,y)=(x-4,y). If x<=4 but y>4, f(x,y)=(x,y-4). Otherwise, f(x,y)=(x+5,y+6). A robot starts by moving to the point (1,1). Every time it arrives at a point (x,y), it applies f to that point and then moves to f(x,y). If the robot runs forever, how many different points will it visit? Thank you.
1,1
6,7
2,7
2,3
7,9
3,9
3,5
3,1
8,7
4,7
4,3
9,6
5,6
1,6
1,2
6,8
2,8
2,4
7,10
3,10
3,6
3,2 this thing is a pretty good vacuum cleaner, keep going until it repeats a point :)
To determine the number of different points the robot will visit, we need to analyze the movement of the robot based on the given function f(x, y).
The robot starts at the point (1, 1). Let's follow the steps to determine its subsequent movements:
1. Since x = 1 and y = 1, neither x > 4 nor y > 4 is true. Therefore, f(1, 1) = (1 + 5, 1 + 6) = (6, 7).
2. Now, the new point is (6, 7). Since x > 4 is true, f(6, 7) = (6 - 4, 7) = (2, 7).
3. The robot moves to (2, 7). Neither x > 4 nor y > 4 is true. Therefore, f(2, 7) = (2 + 5, 7 + 6) = (7, 13).
4. Now, the new point is (7, 13). Since x > 4 is true, f(7, 13) = (7 - 4, 13) = (3, 13).
5. The robot moves to (3, 13). Neither x > 4 nor y > 4 is true. Therefore, f(3, 13) = (3 + 5, 13 + 6) = (8, 19).
6. Now, the new point is (8, 19). Since x > 4 is true, f(8, 19) = (8 - 4, 19) = (4, 19).
From this point on, the robot will keep visiting (4, 19) repeatedly since both x and y are no longer greater than 4. Therefore, the point (4, 19) will be the only point the robot visits infinitely.
To summarize, the robot will visit a total of two different points – (1, 1) and (4, 19).