Let f and g be real - valued functions such that `f(x+y)+f(x-y)=2f(x)*g(y), forall " x , y " in R.` Prove that , if f(x) is not identically zero and `abs(f(x)) le 1, forall x in R,` then `abs(g(y)) le 1, forall y in R.`

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

Let f be differentiable function satisfying f((x)/(y))=f(x) - f(y)"for all" x, y gt 0 . If f'(1) = 1, then f(x) is

If f and g are two real valued functions defined as f(x)= 2x+1 and g(x)= x^(2)+1 , then find fg

Let f be a function satisfying f(x+y)=f(x) + f(y) for all x,y in R . If f (1)= k then f(n), n in N is equal to

Let f(x)=1/2[f(xy)+f(x/y)] " for " x,y in R^(+) such that f(1)=0,f'(1)=2. f(e) is equal to

If f and g are two real valued functions defined as f(x)= 2x+1 and g(x)= x^(2)+1 , then find f+g

If f and g are two real valued functions defined as f(x)= 2x+1 and g(x)= x^(2)+1 , then find f-g

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(x)=1/2[f(xy)+f(x/y)] " for " x,y in R^(+) such that f(1)=0,f'(1)=2. f(x)-f(y) is equal to

Let f(x)=1/2[f(xy)+f(x/y)] " for " x,y in R^(+) such that f(1)=0,f'(1)=2. f'(3) is equal to