f-x-y-f-x-y-xy-f-x-y-x-2-y-2-f-x-y-y-x-100-y-100-f-0-0-f-1-4-

Question Number 1842 by 123456 last updated on 12/Oct/15

f (x, y) = f (x + y, x y)

f (x, y) = x, - 2 ⩽ y ⩽ 2

f (x, y) = y, ∣ x ∣⩾ 100 \lor ∣ y ∣⩾ 100

f (0, 0) = ?

f (1, 4) = ?

Commented by 123456 last updated on 12/Oct/15

(1, 4) \to (5, 4) \to (9, 20) \to (29, 180) \to \dots

(x, x) \to (2 x, x^{2}) \to (2 x + x^{2}, 2 x^{3}) \to \dots

Answered by Rasheed Soomro last updated on 13/Oct/15

f (0, 0) = 0 [∵ f (x, y) = x, - 2 ⩽ y ⩽ 2 a n d h e r e x = 0]

S i n c e f (x, y) = f (x + y, x y)

T h e r e f o r e f (1, 4) = f (1 + 4, 1.4) = f (5, 4)

= f (9, 20) = f (29, 180), h e r e ∣ y ∣⩾ 100

f (29, 180) = 180 [∵ f (x, y) = y, ∣ x ∣⩾ 100 \lor ∣ y ∣⩾ 100]

S o,

f (1, 4) = f (29, 180) = 180

C o n t i n u e i n g f u r t h e r i n t h i s w a y w e c a n a l s o f i n d

o t h e r v a l u e s o f f (1, 4)

f (1, 4) = f (29, 180) = f (209, 5220) = 5220

f (1, 4) = 180, 5220, \dots