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 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+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 f(x)=a x+b , where a ,\ b are constants and a >0 is an increasing function on R .

Prove that f(x)=a x+b , where a , b are constants and a >0 is an increasing function on Rdot

Prove that f(x)=a x+b , where a ,\ b are constants and a<0 is a decreasing function on R .

If f(a+b+1-x)=f(x) , for all x where a and b are fixed positive real numbers, the (1)/(a+b) int_(a)^(b) x(f(x)+f(x+1) dx is equal to :

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 even function and f _ 2 (x) is an odd function. Then f _ 1 ( x + y ) + f _ 1 ( x - y ) equals :

if a,b are two positive numbers 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 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 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.