Prove that `f(x)gi v e nb yf(x+y)=f(x)+f(y)AAx in R` is an odd 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+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.

Prove that the function f given by f(x)= |x-1|, x in R is not differentiable at x= 1.

Which of the following functions have the graph symmetrical about the origin? (a) f(x) given by f(x)+f(y)=f((x+y)/(1-x y)) (b) f(x) given by f(x)+f(y)=f(xsqrt(1-y^2)+ysqrt(1-x^2)) (c) f(x) given by f(x+y)=f(x)+f(y)AAx , y in R (d) none of these

Suppose the function f(x) satisfies the relation f(x+y^3)=f(x)+f(y^3)dotAAx ,y in R and is differentiable for all xdot Statement 1: If f^(prime)(2)=a ,t h e nf^(prime)(-2)=a Statement 2: f(x) is an odd function.

Let f(x)=a^(x)(a gt 0) be written as f(x)=f_(1)(x)+f_(2)(x), " where " f_(1)(x) is an function and f_(2)(x) is an odd function. Then f_(1)(x+y)+f_(1)(x-y) equals

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:R to R be defined by f(x) =e^(x)-e^(-x). Prove that f(x) is invertible. Also find the inverse function.

Consider a real-valued function f(x) satisfying 2f(x y)=(f(x))^y+(f(y))^xAAx , y in Ra n df(1)=a ,w h e r ea!=1. Prove that (a-1) sum_(i=1)^nf(i)=a^(n+1)-a

Statement 1: If f(x) is an odd function, then f^(prime)(x) is an even function. Statement 2: If f^(prime)(x) is an even function, then f(x) is an odd function.