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`

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 the function satisfying f^2(x+y)=f^2(x)+f^2(y)AAx ,y in Rdot

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

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

A function f:R rarr R satisfies the equation f(x+y)=f(x)f(y) for all x,y in R , f(x) ne 0 . Suppose that the function is differentiable at x=0 and f'(0)=2. Prove that f'(x)=2f(x).

f(x) is a polynomial function, f: R rarr R, such that f(2x)=f'(x)f''(x). f(3) is

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

A function f : R to R satisfies the equation f(x + y) = f(x), f(y) for all x, y in R , f(x) ne 0 . Suppose that the function is differentiable at x = 0 and f' (0) = 2. Find (f'(x))/f(x)

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

f(x) is real valued function, satisfying f(x+y)+f(x-y)=2f(X),f(y) for all y ne R , then