If `f(x+y)=f(x)dotf(y)` for all real `x , ya n df(0)!=0,` then prove that the function `g(x)=(f(x))/(1+{f(x)}^2)` is an even function.

If f(x) satisfies the relation f(x+y)=f(x)+f(y) for all x , y in Ra n df(1)=5,t h e n (a) f(x)i sa nod dfu n c t ion (b) f(x) is an even function (c) sum_(n=1)^mf(r)=5^(m+1)C_2 (d) sum_(n=1)^mf(r)=(5m(m+2))/3

If f(x) be an even function. Then f'(x)

If f(x+f(y))=f(x)+yAAx ,y in Ra n df(0)=1, then prove that int_0^2f(2-x)dx=2int_0^1f(x)dxdot

If f(a-x)=f(a+x) " and " f(b-x)=f(b+x) for all real x, where a, b (a gt b gt 0) are constants, then prove that f(x) is a periodic function.

If the function / satisfies the relation f(x+y)+f(x-y)=2f(x),f(y)AAx , y in R and f(0)!=0 , then (a) f(x) is an even function (b) f(x) is an odd function (c) If f(2)=a ,then f(-2)=a (d) If f(4)=b ,then f(-4)=-b

If f(x+f(y))=f(x)+yAAx ,y in Ra n df(0)=1, then find the value of f(7)dot

Let f(x) be real valued and differentiable function on R such that f(x+y)=(f(x)+f(y))/(1-f(x)dotf(y)) f(x) is Odd function Even function Odd and even function simultaneously Neither even nor odd

Let f(x+y)=f(x)dotf(y) for all xa n dydot Suppose f(5)=2a n df^(prime)(0)=3. Find f^(prime)(5)dot

Let f(x+y)=f(x)+f(y)+2x y-1 for all real xa n dy and f(x) be a differentiable function. If f^(prime)(0)=cosalpha, then prove that f(x)>0AAx in Rdot

If f(x-y),f(x)f(y),a n df(x+y) are in A.P. for all x , y ,a n df(0)!=0, then (a) f(4)=f(-4) (b) f(2)+f(-2)=0 (c) f^(prime)(4)+f^(prime)(-4)=0 (d) f^(prime)(2)=f^(prime)(-2)