If `F(x)=e^(x)` then prove that `f(x+y)=f(x)*f(y)`

If f(x)=log_(e)x, then prove that :f(xyz)=f(x)+f(y)+f(z)

Let f(x)=((2008)^(x)+(2008)^(-x))/(2),g(x)=((2008)^(x)-(2008)^(-x))/(2) then prove that f(x+y)=f(x)f(y)+g(x)g(y)

If f(x)=log_(e)x and g(x)=e^(x) , then prove that : f(g(x)}=g{f(x)}

A function f:R rarr R satisfies the equation f(x+y)=f(x)f(y) for allx,y in R and f(x)!=0 for all x in R .If f(x) is differentiable at x=0 .If f(x)=2, then prove that f'(x)=2f(x) .

If y=f(x)=(ax-b)/(bx-a) , the prove that : x=f(y)

If y=f(x)=(3x+1)/(5x-3) , prove that x=f(y).

For a function f(x),f(0)=0 and f'(x)=(1)/(1+x^(2)), prove that f(x)+f(y)=f((x+y)/(1-xy))

A function f is defined such that for all real x,y(a)f(x+y)=f(x).f(y)(b)f(x)=1+xg(x) where lim_(x rarr0)g(x)=1 prove that f;(x)=f(x) and f(x)=e^(x)

If f(x+y)=f(x)+f(y),AAx ,\ y in R then prove that f(k x)=kf(x)for\ AAk ,\ x in Rdot

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