Even and Odd Functions

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Introduction to Even and Odd Functions

Functions are the basis of calculus. It is fundamental to the understanding of any real-life application of applied mathematics. A function basically indicates a correspondence between two variables. The dependence can be established in different forms such as tabular form, graphs and charts, or equational form.

To define, function from X to Y is a rule that maps each element in set X with a unique element in set Y. Set X is called the domain of the function whereas set Y is termed as co-domain.


Variable in Functions

If f is a function, it is denoted as y = f(x); x is called the independent/exogenous variable and y is called the dependent/endogenous variable.

For a function to be defined, every element in the domain should be mapped to a unique element in co-domain. However, not every element in the co-domain may be related to an element of the domain. The specific value of f(x) is known as the image of element x by function f. Therefore, the set of all elements in set Y that are images to at least one element in set X is called the range of the function. Range is a subset of co-domain as shown in the image below.


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Odd functions

If f is a real-valued function on a real set, f is even if:


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-f(x) =f (-x)

Or, f (-x) +f(x) =0

If any given function follows the above rule, it is said to be an odd function.

The graph of any even function is rotationally symmetric along the origin.


Even functions

If f is a real-valued function on a real set, f is even if:

F(x)=f(-x)

Or, f(x)-f (-x) =0

If any given function follows the above rule, it is said to be an even function.

The graph of any even function is symmetric to the y-axis, i.e. it forms a mirror image.


Solved Examples

Some illustrations on how to know if a function is even:

Example 1

F(x) = x² + 1

Solution:

Replacing x with (-x) in f(x)

Therefore, F (-x) - (- x²) + 1

Or, F(-x) = x² + 1

Or, F (-x) = f (x)

Hence, f(x) is an even function.


Example 2

f(x)= cos x

Solution:

Trigonometry confirms us that cos(x) = cos(-x)

Therefore, F (-x) = f (-x)

The cosine function is even. 


Example 3

F(-x) = x3 - x

Now, substituting the value of x with (-x) in f(x)

Therefore, F (-x) = (-x3) - (-x)

Or,

F (-x) = (-x3) + x

Likewise,

-    F (x) = -(x3) – x

-    F (x) = -x3 + x

Thus, it appears that

f(-x) = f(x)

Hence, f(x) is an odd function.


Some Basic Properties Even Odd Functions:

As a rule of thumb, every real-values function can be decomposed using an even and odd function. Let fe (x) represent an even function while fo(x) denotes odd function. Thus,

Any even function fe(x) = [f(x) + f (-x)]/2 and

Every odd function fo(x) = [f(x) - f (-x)]/2 and

 f(x) = fe(x) + fo(x)

And, f(x) = fe(x) + fo(x)

There are instances of some functions satisfying the conditions of both even and odd functions. Such functions are defined everywhere in the real-value set.

The absolute value of an odd function is even.


Algebraic Properties Even Odd Functions

  • The addition of two even functions produces an even function.

  • The addition of two odd functions produces an odd function.

  • The subtraction of two even functions is even.

  • The subtraction of two odd functions is a function.

  • The addition/ difference of even and odd is neither even nor odd, except for the cases where one function is zero. The product of two even functions is even.

  • The multiplication of two odd functions will turn out to be an even function.

  • The multiplication of two even functions will turn out to be an odd function.

  • The division of two even functions is even.

  • The division of two odd functions is an even function.

  • The product/division of an even and odd function is an odd function.

FAQ (Frequently Asked Questions)

Q1. How to Determine if the Function is Odd, Even or Neither?

Answer: Given the formula for a function, we can identify if the function is odd, even or neither. Find below the provision that satiates the verification of the function:

  • Identify if the function satisfies the equation i.e. f(x) = −f(−x)f(x) = −f(−x). If it does, then it is an odd function.

  • Identify if the function satisfies the equation i.e. f(x) = f(−x)f(x) = f(−x). If it does, then it is an even function.

  • If the function does not fulfil the criterion of either rule, it is neither odd nor even.

Q2. What do we Understand by Even and Odd Functions?

Answer: In the mathematical domain, odd functions and even functions are those functions that satisfy specific symmetry relations, in regard to taking additive inverses.

Q3. What Makes Even and Odd Functions Important in Maths?

Answer: Odd functions and even functions are quite important in many areas of mathematical assessment, particularly the concept of Fourier series and power series. They are named with respect to the parity of the powers of the power functions which further satisfy each condition. For example, f(x) = xn is an even function when n is an even integer, and it is an odd function when n is an odd integer.