A Short Note on Odd and Even Function

There are various sorts of functions that we learn in mathematics. We can use algebra or graphics to determine whether a function is even or odd. Even and odd functions can be tested by substituting negative inputs (-x) for x in the function f(x) and examining the resulting output value. The symmetry relations of even and odd functions are used to classify them. The power function f(x) = xn is an even function if n is even, and an odd function if n is odd, hence even and odd functions are named accordingly. A function can be even, odd, or both, or neither even nor odd. 

Even and Odd Functions

A real-valued function is either even or odd in most cases. To test whether a function is even or odd, we substitute -x for x in the function f(x), checking the output value of f(-x) to ascertain the function type. Even and odd functions have the same symmetry. 

Even Function

For all values of x in the domain of f, a real-valued function f(x) is said to be an even function if the output value of f(-x) is the same as f(x). The following equation should be held by an even function: For all values of x in D(f), f(-x) = f(x), where D(f) signifies the domain of the function f. To put it another way, the equation f(-x) – f(x) = 0 holds for every x given an even function. Consider the following example: f(x) = x².

For all values of x, f(-x) = (-x)² = x², since the square of a negative number is the same as the square of its positive value. For all x, this implies f(-x) = f(x). f(x) = x² is hence an even function. Functions like x⁴, x⁶, x⁸, and so on are also functions. 

Odd function

For all values of x in the domain of f, a real-valued function f(x) is said to be an odd function if the output value of f(-x) is the same as the negative of f(x). The following equation should be stored in an odd function: For all values of x in D(f), f(-x) = -f(x), where D(f) signifies the domain of the function f. To put it another way, the equation f(-x) + f(x) = 0 holds for every x for an odd function. Consider the following example: f(x) = x³.

Because the cube of a negative number is the same as the negative of the cube of the positive value of the number, f(-x) = (-x)³ = -(x³) for all values of x. For all x, this implies f(-x) = -f(x). As a result, f(x) = x³ is an unusual function. Similarly, functions like x⁵, x⁷, x⁹, and so on are unusual. 

Both Even and Odd Function

If a real-valued function f(x) satisfies f(-x) = f(x) and f(-x) = -f(x) for all values of x in the domain of the function f, it is said to be both even and odd (x). The only function that is both even and odd is the zero function, which has f(x) = 0 for every x. For all values of x, we know that f(-x) = -f(x) = f(x) = 0 for the zero function. As a result, f(x) = 0 is both an even and an odd function. 

Neither Even nor Odd Function

It is claimed that a real-valued function f(x) is neither even nor odd if it does not satisfy the conditions f(-x) = f(x) and f(-x) = -f(x) for at least one value of x in the domain of the function f(x), respectively (x). Let us look at an example to better grasp what the definition means. Consider the functions f(x) = 2x⁵ + 3x² + 1, f(-x) = 2(-x)⁵ + 3(-x)² + 1 = -2x⁵ + 3x² + 1, which are neither equal to f(x) nor -f(x). As a result, the function f(x) = 2x⁵ + 3x² + 1 is neither an even nor an odd function. 

Properties of Even and Odd Functions

• The sum of two even functions is equal to two even functions, whereas the sum of two odd functions is equal to two odd functions.

• The difference between two even functions is equal to one, and the difference between two odd functions is equal to two odd functions.

• Unless one of the functions is a zero function, the sum of an even and an odd function is neither even nor odd in any case.

• It is also true that the product of two equal functions is equal, and that the product of two odd functions is equal to the product of two even functions.

• When an even and an odd function are combined, the result is odd.

• The quotient of two even functions is equal to the product of the two odd functions, and the product of the two even functions is equal to the product of the two odd functions.

• A function that is the product of an even and an odd function is called an odd function.

• In the case of two even functions, the result is even, whereas the result of two odd functions is odd, and vice versa.

• Even functions are formed by combining an even and an odd function. 

Example of Even and Odd Functions

E.g. 1. Identify whether the function f(x) = sinx.cosx is an even or odd function. Verify using the even and odd functions definition. 

Solution: Given function f(x) = sinx.cosx. 

Sinx is an odd function, whereas cosx is an even function. In addition, the product of two even functions is odd. As a result, f(x) = sinx. cosx is a peculiar function. Let’s see if we can prove it with the definition of an odd function. 

Consider f(-x) = sin(-x).cos(-x) = -sinx.cosx = -f(x). Therefore, f(x) is an odd function. Hence, verified. 

Conclusion

Even functions and odd functions are mathematical functions that meet specific symmetry relations when taking additive inverses. They are crucial in many fields of mathematical analysis, particularly in the theory of power and Fourier series.