Question

Define odd function and even function.

Answer 1

Hint: As this is a theoretical problem we have to be thorough with the definition. If we considered function as $$f(x)$$ and if you substitute in the function based on the solution obtained present area even or odd function. Based on symmetry we can say which is even function and which is odd function.

Complete step-by-step solution -
Even function: A function $$f$$ defined on$$[-a,a]$$ is to be an even function if $$f(-x)=f\left(x\right)$$$$\mathrm{\forall }$$$$x\in [-a,a]$$
Example for even function is $$f\left(x\right)=x^2$$
 Odd function: A function $$f$$ defined on $$[-a,a]$$ is to be an odd function if $$f(-x)=-f\left(x\right)$$$$\mathrm{\forall }$$$$x\in [-a,a]$$
Example for odd function is $$f\left(x\right)=x^3$$
The only function whose domain is all real numbers which is both odd and even, is the constant function which is identically zero, f(x)=0.
The sum of two even functions is even, and the sum of two odd functions is odd.
The difference of two even functions is even, and the difference of two odd functions is odd.
The product of two even functions is even, and the product of two odd functions is even.
The product of an even function and an odd function is an odd function.
The quotient of two even functions is even, and the quotient of two odd functions is even.
The quotient of an even function and an odd function is an odd function.
The derivative of an even function is odd, and the derivative of an odd function is even.
The composition of two even functions is even, and the composition of two odd functions is odd.
The composition of an even function and an odd function is even.

Note: This is a direct problem we have to solve this by knowing the definition. Based on the symmetry we can include that which type of function either even function or odd function. Properties are also mentioned above which defines the complete nature of Even and Odd functions.