Even and Odd Functions Explained with Graphs and Properties

Even and Odd functions form an important aspect in several mathematical analyses. These are functions which when taking an additive inverse, satisfy a specific symmetry.  A basic understanding of these functions is crucial for someone who wants to apply Calculus to real-life situations. These functions have several important applications in Mathematics.  Odd and even functions can be depicted algebraically or in the form of graphs. Refer to the official website of Vedantu for an elaborate and comprehensive explanation.

Functions are the basis of calculus. It is fundamental to the understanding of any real-life application of applied mathematics. A function 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 the 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. The range is a subset of co-domain as shown in the image below.

Odd 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 odd function.

The graph of any even function is rotationally symmetric along with 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

It is essential to go through some examples to understand the above-mentioned points properly. 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 of 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)f(x)+f(−x)/2 and

Every odd function fo(x) = f(x)−f(−x)f(x)−f(−x)/2 and

 f(x) = f_e(x) + f_o(x)

And, f(x) = f_e(x) + f_o(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.

Practical tips to master the concepts of odd and even functions:

Even and odd functions form part of usual calculus. Those who find the concepts difficult to master can follow these simple tips to excel in the subject:

• Understand the meaning of even and odd functions by going through the definitions.

• Carefully go through the properties of each function multiple times.

• Try to refer to multiple sources to study the same topic to get a better overall perspective

• Use Vedantu's solutions when solving questions from the topic to understand the ideal way to reach the solutions.

• Practice as many questions as possible from the topic. Calculus and all the other seemingly difficult concepts become easier with diligent practice. Draw graphs for each question.

• Understand the practical applications of odd and even functions. Understanding the objective behind learning a particular topic helps to better relate to the concepts and understand them in a more profound way.