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how can we find out f(x)?? if something like f(x+y)=f(x)+f(y) is given and it also say f(1)=2 so now also find f(2)..?? for functional equations watz the procedure to find out f(x)??

how can we find out f(x)?? if something like f(x+y)=f(x)+f(y) is given and it also say f(1)=2 so now also find f(2)..??  for functional equations watz the procedure to find out f(x)??

Grade:12

2 Answers

Vanya Saxena
18 Points
13 years ago

Hi,

It is given that f(x+y) =f(x)+f(y)  and f(1)=2

      so on keeping x=1 and y=0 in the above eqn we  get

                                  

                                                        f(1)=f(1) +f(0)  =>  f(0)=0  now when we keep x=1 and y=1 we get

  

                                                          f(2)=f(1)+f(1)  =>f(2)=4

 

similarly when we calculate f(3) and f(4) we get them as 6 and 8 respectively.On observing the sequence we come to know that

                                                                      f(x)=2x

 

 

Ramesh V
70 Points
13 years ago

put x=y

then f(2x)=2f(x)

so f(0) = 0

given f(1) = 2

so for x=y=1 f(2) = 2f(1) = 4

so f(2) =4

for x= -y

f(x) + f(-x) = 0 

so f(-x) = -f(x)  so f(x) is odd function

if we take f(x) to be a polynomia function then

f(x) is of form (px)2n+1

for p=2 and n=0 , we have f(x) = 2x    which fits in f(0) , f(1) and f(2) .

SO f(x) = 2x

 

 

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