Linear functional calculation

Algebra Level 3

f ( x , y ) = x + ( y × f ( y , x ) ) f(x,y) = x+ (y \times f(y,x))

Find f ( 1 , 2 ) f(1,2) .


The answer is -5.

অস্পষ্ট দর্শনানুসারি by অস্পষ্ট দর্শনানুসারি , 18, Bangladesh

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3 solutions

Sabhrant Sachan
May 1, 2016

Put x = 1 and y = 2 ⟹ f ( 1 , 2 ) = 1 + 2 × f ( 2 , 1 ) Now put x = 2 and y = 1 ⟹ f ( 2 , 1 ) = 2 + 1 × f ( 1 , 2 ) Substitute the value of f ( 2 , 1 ) in the 1st Equation, we get ⟹ f ( 1 , 2 ) = 1 + 2 ( 2 + f ( 1 , 2 ) ) = f ( 1 , 2 ) = 1 + 4 + 2 f ( 1 , 2 ) = f ( 1 , 2 ) = − 5 \text {Put } x=1 \text { and } y=2 \\ \implies f(1,2)=1+2\times f(2,1) \\ \text {Now put } x=2 \text { and } y=1 \\ \implies f(2,1)=2+1\times f(1,2) \\ \text {Substitute the value of } f(2,1) \text { in the 1st Equation, we get } \\ \implies f(1,2)=1+2(2+f(1,2)) \\ = f(1,2)=1+4+2f(1,2) \\ = \color{#3D99F6}{\boxed {f(1,2)=-5}}

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By the eqn f(1,2)=1+2f(2,1) or 2f(2,1)=f(1,2)-1 again 2f(2,1)=4+2f(1,2) so , by these two eqn 4+2f(1,2)-f(1,2)+1 then f(1,2)= -5

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Substituting ( 1 , 2 ) (1,2) and ( 2 , 1 ) (2,1) gives a system of linear equations in two variables which can easily be solved.

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Pranshu Gaba - 5 years, 1 month ago

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Thanks it was very helpful to me............

অস্পষ্ট দর্শনানুসারি - 5 years, 1 month ago
Mansi Khandelwal
Jan 16, 2017

this is wrong ques because f(1,1) gives 1=0

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