A function `f(x)` is so defined that for all real `x,{f(x)}^n = f(nx).` Prove that `f(x) · f '(nx) = f'(x) · f(nx).`

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)

A function f(x) is defined for all real x and satisfied f(x+y)=f(xy),AA x,y. If f(1)=-1 then f(2006) equals

The function f(x) is defined for all real x .If f(x+y)=f((xy)/(4))AA x y and f(-4)=-4 then |f(2016)| is

The function fis not defined for =0, but for all non zero real number x,f(x)+2f((1)/(x))=3x. The equation f(x)=f(-x) is satisfied by

Given a differentiable function f(x) defined for all real x, and is such that f(x+h)-f(x)<=6h^(2) for all real h and x . Show that f(x) is constant.

If f: R to [0,∞) be a function such that f(x -1) + f(x + 1) = sqrt3(f(x)) then prove that f(x + 12) = f(x) .

A function f: R to R is defined as f(x)=4x-1, x in R, then prove that f is one - one.

Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f '(x) ne 0 for all x in R . If |[f(x)" "f'(x)], [f'(x)" "f''(x)]|= 0 , for all x in R , then the value of f(1) lies in the interval:

Let f(x+y)=f(x)+f(y) for all real x,y and f'(0) exists.Prove that f'(x)=f'(0) for all x in R and 2f(x)=xf(2)

Let (x) is a real function, defines as f(x) =(x-1)/(x+1), then prove that f(2x)=(3f(x)+1)/(f(x)+3).