Functional Equation
If f is a function satisfying f ( x + y ) = 3 y f ( x ) + 2 x f ( y ) for all x , y ∈ R and f ( 1 ) = 1 , what is the value of f ( 3 ) ?
The answer is 19.
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4 solutions
very nice method. i like it but i solve it different method .
All we simply do is compute f ( 2 ) and use that for f ( 3 )
f ( 2 ) = f ( 1 + 1 ) = 3 f ( 1 ) + 2 f ( 1 ) = 5 f ( 1 ) = 5
f ( 3 ) = f ( 2 + 1 ) = 3 f ( 2 ) + 4 f ( 1 ) = 1 9
and we are done.
As a side note by inspection f ( x ) = 3 x − 2 x
I did it the same way. I'm wondering why is this level 3 when it may very well be 2.
please tell me complete solution with details i can't satisfy completely.
Great fun! Bears a strong analogy to the factorial function, as this code displays. Incidentally, f(10)=58025, just in case anyone wants to know.
Love to see someone using python :-) Kudos points :-)
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Thank you. Like a tank to crack a walnut in this case.
first we compute f(2) and go for f(3)
given f(1)=1
f(2)=f(1+1)=3+2=5
f(3)=f(2+1)=19
First we substitute ( x ; y ) → ( x ; 1 ) . This gives us: f ( x + 1 ) = 3 f ( x ) + 2 x ⇔ f ( x ) = 3 f ( x − 1 ) + 2 x − 1 Next we define a new function g ( x ) such that: g ( x ) = f ( x ) + 2 x = 3 f ( x − 1 ) + 2 x − 1 + 2 . 2 x − 1 = 3 ( f ( x − 1 ) + 2 x − 1 ) = 3 g ( x − 1 ) This implies that for integer x g ( x ) is a geometric progression and g ( x ) = g ( 1 ) . 3 x − 1 . This can be extended to all real numbers. We are given f ( 1 ) = 1 and by the definition of g ( x ) we have g ( 1 ) = f ( 1 ) + 2 = 3 and so g ( x ) = 3 x . From the definition of g ( x ) we find: f ( x ) = g ( x ) − 2 x = 3 x − 2 x Therefore f ( 3 ) = 1 9 and this is our answer.
I know that @Ishan Tarunesh has a better and a more straightforward solution, but this is how I did it :)
f ( x + y ) = 3 y f ( x ) + 2 x f ( y )
Replace x and y with each other
f ( y + x ) = 3 x f ( y ) + 2 y f ( x )
Since LHS is same in both case, RHS must also be equal
2 x f ( y ) + 3 y f ( x ) = 3 x f ( y ) + 2 y f ( x )
Rearranging we get 3 x f ( y ) − 2 x f ( y ) = 3 y f ( x ) − 2 y f ( x )
Put y = 1 we get
f ( x ) = 3 x − 2 x
Now f(3) = 19