If a function `f: R ->R` be such that `f(x-f(y)) = f(f(y) )+xf(y)+f(x) -1 AA x , y in R` then `f(2)=`

Determine all functions f : R rarr R satisfying f(x-f(y)) = f(f(y)) + xf(y) + f(x) - 1 AA x, y in R

Let f be a polynomial function such that f(x) f(y) + 2 = f(x) + f(y) + f(xy) AA x,y in R^+ cup {0} and f(x) is one-one AA x,y in R^+ with f(0)=1 and f(1)=2 Answer the following questions based on above passage: The function y=f(x) is given by

Let f be a function such that f(x + y) = f(x) + f(y) forall x , y in R , if f(1) = k, then show that f(n) is equal to nk.

A function f: R->R satisfies the equation f(x+y)=f(x)f(y) for all x , y in R and f(x)!=0 for all x in Rdot If f(x) is differentiable at x=0a n df^(prime)(0)=2, then prove that f^(prime)(x)=2f(x)dot

A function f: R -> R satisfy the equation f (x)f(y) - f (xy)= x+y for all x, y in R and f(y) > 0 , then

Consider a differentiable function f:R->R for which f'(0)=ln2 and f(x + y) = 2^x f(y) + 4^y f(x) AA x, y in R which of the following is /are correct? (a) f(4)=240 (b) f prime (2)=24 ln 2 (c) The minimum value of y=f(x) is -1/4 (d) the number of solutions of f(x)=2 is 1

If a function satisfies (x-y) f(x+y)-(x+y) f(x-y) = 2(x^2y - y^3), AA x, y in R and f(1) = 2, then

If the function f satisfies the relation f(x+y)+f(x-y)=2f(x)xxf(y), AA x, y, in R and f(0) ne 0 , then

If f(x + y) = f(x) - f(y), forall x, y in R , then show that f(3) = f(1)

Consider a differentiable f:R to R for which f(1)=2 and f(x+y)=2^(x)f(y)+4^(y)f(x) AA x , y in R. The value of f(2) is