Let `f(x)` be periodic and `k` be a positive real number such that `f(x + k) + f (x) = 0` for all `x in R`. Then the period of `f(x)` is

If f(x)> x ;AA x in R. Then the equation f(f(x))-x=0, has

Let f : N to N : f(x) =2 x for all x in N Show that f is one -one and into.

Let f(x) be a real valued function such that f(0)=1/2 and f(x+y)=f(x)f(a-y)+f(y)f(a-x), forall x,y in R , then for some real a, (a)f(x) is a periodic function (b)f(x) is a constant function (c) f(x)=1/2 (d) f(x)=(cosx)/2

Let f be a function such that f(x+f(y)) = f(x) + y, AA x, y in R , then find f(0). If it is given that there exists a positive real delta such that f(h) = h for 0 lt h lt delta , then find f'(x)

Let f(x) be a function such that : f(x - 1) + f(x + 1) = sqrt3 f(x), for all x epsilon R. If f(5) = 100, then prove that the value of sum_(r=o)^99 f(5 + 12r) will be equal to 10000.

Find period of f(x)=tan3x+sin(x/3) .

Let f:R->R be a function such that f(x+y)=f(x)+f(y),AA x, y in R.

Let f(x) and g(x) be functions which take integers as arguments. Let f(x + y) =f(x)+ g(y) + 8 for all intege x and y. Let f(x) = x for all negative integers x and let g (8) = 17 . Find f(0).

Let f be a differentiable function satisfying [f(x)]^(n)=f(nx)" for all "x inR. Then, f'(x)f(nx)

Let f(x, y) be a periodic function satisfying f(x, y) = f(2x + 2y, 2y-2x) for all x, y; Define g(x) = f(2^x,0) . Show that g(x) is a periodic function with period 12.