If `f(x)` satisfies the relation `f(x)+f(x+4)=f(x+2)+f(x+6)` for all`x ,` then prove that `f(x)` is periodic and find its period.

If f (x) satisfies the relation 2 f(x ) + f(1-x) =x^2 for all real x, then f (x) is

If f: Rrarr[0,oo) is a function such that f(x-1)+f(x+1)=sqrt(3)f(x), then prove that f(x) is periodic and find its period.

If a , b are two fixed positive integers such that f(a+x)=b+[b^3+1-3b^2f(x)+3b{f(x)}^2-{f(x)}^3]^(1/3) for all real x , then prove that f(x) is periodic and find its period.

If function f satisfies the relation f(x)*f^(prime)(-x)=f(-x)*f^(prime)(x) for all x ,

If f : RR to RR is defined by f (x) = 2x -3, then prove that f is a bijection and find its inverse.

Let f(x, y) be a periodic function satisfying f(x, y) = f(2x + 2y, 2y-2x) for all x, y; Define g(x) = f(2^x,0) . Show that g(x) is a periodic function with period 12.

If f:R to R is defined by f(x) = 3x - 5 , prove that f is a bijection and find its inverse .

H-7.Let'f' be a real valued function defined for all real numbers x such that for some positive constant 'a' theequation f(x+a)= 2 1 ​ + f(x)−(f(x)) 2 holds for all x. Prove that the function f is periodic.

y=f(x) satisfies the relation int_(2)^(x)f(t)dt=(x^(2))/2+int_(x)^(2)t^(2)f(t)dt The range of y=f(x) is

y=f(x) satisfies the relation int_(2)^(x)f(t)dt=(x^(2))/2+int_(x)^(2)t^(2)f(t)dt The range of y=f(x) is