Show that any function can be expressed as the sum of an odd function and even function

Statement-1: Every function can be uniquely expressed as the sum of an even function and an odd function. Statement-2: The set of values of parameter a for which the functions f(x) defined as f(x)=tan(sinx)+[(x^(2))/(a)] on the set [-3,3] is an odd function is , (9,oo)

Prove that the derivative of an even function is an odd function and that of an odd function is an even function.

Statement I Integral of an even function is not always an odd function. Statement II Integral of an odd function is an even function .

Which of the following function is an even function ?

Left hand derivative and right hand derivative of a function f(x) at a point x=a are defined as f'(a^-)=lim_(hrarr0^(+))(f(a)-f(a-h))/(h) =lim_(hrarr0^(+))(f(a+h)-f(a))/(h) andf'(a^(+))=lim_(hrarr0^(+))(f(a+h)-f(a))/(h) =lim_(hrarr0^(+))(f(a)-f(a+h))/(h) =lim_(hrarr0^(+)) (f(a)-f(x))/(a-x) respectively. Let f be a twice differentiable function. We also know that derivative of a even function is odd function and derivative of an odd function is even function. If f is even function, which of the following is right hand derivative of f' at x=a?

Let G(x)=(1/(a^x-1)+1/2)F(x), where a is a positive real number not equal to 1 and f(x) is an odd function. Which of the following statements is true? G(x) is an odd function G(x)i s an even function G(x) is neither even nor odd function. Whether G(x) is an odd or even function depends on the value of a

Find the sum and product of identity function and the modulus function.

f(x)=(cosx)/([(2x)/pi]+1/2), where x is not an integral multiple of pi and [dot] denotes the greatest integer function, is an odd function an even function neither odd nor even none of these