f-x-y-f-x-1-y-y-x-gt-0-f-x-y-y-1-x-x-0-y-gt-0-xy-x-0-y-0-f-5-7-f-6-9-

Question Number 3793 by 123456 last updated on 21/Dec/15

f (x, y) = {\begin{cases} f (x - 1, y) + y & x > 0 \\ f (x + y, y - 1) + x & x ⩽ 0 \land y > 0 \\ x y & x ⩽ 0 \land y ⩽ 0 \end{cases}

f (5, 7) = ?

f (6, 9) = ? ?

Commented by prakash jain last updated on 21/Dec/15

y > 0

f (0, y) = f (y, y - 1) + 0

= f (0, y - 1) + (y - 1) y

= f (0, y - 2) + (y - 2) (y - 1) + (y - 1) y

= f (0, y - 3) + (y - 3) (y - 2) + (y - 2) (y - 1) + (y - 1) y

f (0, y) = f (0, 0) + 1 \cdot 2 + 2 \cdot 3 + \dots + (y - 1) y

\underset{n = 1}{\sum^{y - 1}} n (n + 1) = \frac{(y - 1) (y) (2 y - 1)}{6} + \frac{y (y - 1)}{2}

f (0, y) = \frac{y (y - 1) (2 y - 1)}{6} + \frac{y (y - 1)}{2} (y > 0)

f (x, y) = f (0, y) + x y

f (x, y) = \frac{y (y - 1) (2 y - 1)}{6} + \frac{y (y - 1)}{2} + x y (x > 0, y > 0)

f (5, 7) = f (0, 7) + 35

= \frac{7 \times 6 \times 13}{6} + \frac{7 \times 6}{2} + 35

= 91 + 21 + 35

= 112 + 35 = 147

f (6, 9) = f (0, 9) + 6 \times 9

= \frac{9 \times 8 \times 17}{6} + \frac{9 \times 8}{2} + 54

= 3 \times 4 \times 17 + 36 + 54

= 12 \times 17 + 90

= 204 + 90 = 294

Commented by Rasheed Soomro last updated on 22/Dec/15

\in xcellent for deriving a formula

for f (x, y) in terms of x and y only (f r e e o f

a n y function definition)!

Answered by Yozzii last updated on 21/Dec/15

f (5, 7) \Rightarrow x > 0 (y > 0) .

∴ f (5, 7) = f (4, 7) + 7

= {f (3, 7) + 7} + 7

= {f (2, 7) + 7} + 14

= {f (1, 7) + 7} + 21

= {f (0, 7) + 7} + 28

f (0, 7) \Rightarrow x ⩽ 0 \land y > 0 .

∴ f (5, 7) = {f (0 + 7, 7 - 1) + 0} + 35

= f (7, 6) + 35

= f (6, 6) + 6 + 35

W e s e e t h a t f (x, y) = f (0, y) + x y i f

x > 0 (\forall y) . T h i s i s s e e n f r o m c o m p u t a t i o n

o f f (5, 7) = f (0, 7) + 5 \times 7 .

∴ f (7, 6) = f (0, 6) + 6 \times 7 = 42 .

∴ f (5, 7) = f (0, 6) + 6 \times 7 + 5 \times 7

= f (6, 5) + 6 \times 7 + 5 \times 7

= f (0, 5) + 6 \times 5 + 6 \times 7 + 5 \times 7

= f (5, 4) + 30 + 42 + 35

= f (0, 4) + 20 + 72 + 35

= f (4, 3) + 127

= f (0, 3) + 12 + 127

= f (3, 2) + 139

= f (0, 2) + 6 + 139

= f (2, 1) + 145

= f (0, 1) + 2 \times 1 + 145

= f (1, 0) + 147

= f (0, 0) + 0 \times 1 + 147

f (0, 0) \Rightarrow x, y ⩽ 0 \Rightarrow f (0, 0) = 0 .

\Rightarrow f (5, 7) = 147 .

Answered by Yozzii last updated on 21/Dec/15

f (6, 9) = f (5, 9) + 9

= f (4, 9) + 9 + 9

= f (3, 9) + 9 + 9 + 9

= f (2, 9) + 9 + 9 + 9 + 9

= f (1, 9) + 5 \times 9

= f (0, 9) + 6 \times 9

= f (9, 8) + 54

= f (0, 8) + 8 \times 9 + 54

= f (8, 7) + 72 + 54

= f (0, 7) + 8 \times 7 + 126

= f (7, 6) + 56 + 126

= f (0, 6) + 6 \times 7 + 182

= f (6, 5) + 42 + 182

= f (0, 5) + 6 \times 5 + 224

= f (5, 4) + 254

= f (0, 4) + 20 + 254

= f (4, 3) + 274

= f (0, 3) + 12 + 274

= f (3, 2) + 286

= f (0, 2) + 6 + 286

= f (2, 1) + 292

= f (0, 1) + 2 + 292

= f (1, 0) + 294

= f (0, 0) + 0 + 294

= 0 + 294

f (6, 9) = 294

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