FUNCTIONAL EQUATIONS

 

Let us understand this topic from a basic question given below:-

Q. f(x+y) = f(x) + f(y)   , f(3) = 9

find f(2).

Sol. Firstly differentiate above equation with respect to x (assuming y as constant) ,

f '(x+y) = f '(x)   -(1)

Secondly differentiate above equation with respect to y (assuming x as constant) ,

f '(x+y) = f '(y)   -(2)

From equation (1) and (2) we get,

f '(x) = f '(y) = k (say)

Taking f '(x) = k   -(3)

Now, integrate equation (3) with respect to x we get,

f(x) = kx + c     -(4)

Also, we have f(x+y) = f(x) + f(y)   <== put x = y = 0 in it , we get;

f(0+0) = f(0) + f(0)

f(0) = 2 f(0)

f(0) = 0   <== put this in equation (4) we get,

f(0) = k . 0 + c

0 = 0 + c

c = 0    <== put back in equation (4) we get,

f(x) = kx    -(5)

Also f(3) = 9    <== put this in equation (5),

f(3) = k. 3

9 = 3k

k = 3

Hence, f(x) = 3x

f(2) = 3 . 2

f(2) = 6