Key Differences Between Even and Odd Functions with Examples
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) = 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.
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.
FAQs on Even and Odd Functions Explained
1. What is the definition of an even and an odd function?
Even and odd functions are classified based on their symmetry.
An even function is one where substituting -x for x gives the original function. The algebraic rule is f(-x) = f(x). Graphically, an even function is symmetric with respect to the y-axis.
An odd function is one where substituting -x for x gives the negative of the original function. The algebraic rule is f(-x) = -f(x). Graphically, an odd function is symmetric with respect to the origin (180° rotational symmetry).
2. How do you determine if a function is even, odd, or neither?
To determine a function's parity, follow these steps:
1. Given a function f(x), find the expression for f(-x) by replacing every 'x' with '-x'.
2. Simplify the expression for f(-x).
3. Compare the result with the original function f(x):
If f(-x) = f(x), the function is even.
If f(-x) = -f(x), the function is odd.
If f(-x) does not equal f(x) or -f(x), the function is neither even nor odd.
3. What are some common examples of even and odd functions?
Here are some common examples from algebra and trigonometry:
Even Functions: f(x) = x², f(x) = x⁴ (any even power of x), f(x) = |x|, and the trigonometric function f(x) = cos(x).
Odd Functions: f(x) = x, f(x) = x³ (any odd power of x), f(x) = 1/x, and the trigonometric functions f(x) = sin(x) and f(x) = tan(x).
4. Why is an even function symmetric about the y-axis?
The symmetry of an even function about the y-axis is a direct result of its definition, f(x) = f(-x). This equation means that for any x-value, the function has the same y-value as it does for the corresponding negative x-value. For example, if the point (2, 4) is on the graph, the point (-2, 4) must also be on the graph. When you plot these pairs of points, they create a mirror image across the y-axis.
5. How are the properties of even and odd functions used when combining them?
The properties of even and odd functions lead to predictable results when they are combined through arithmetic operations, similar to the rules for even and odd numbers:
The product or quotient of two even functions is even.
The product or quotient of two odd functions is even.
The product or quotient of an even and an odd function is odd.
The sum or difference of two even functions is even, and the sum or difference of two odd functions is odd.
6. Can a function be both even and odd at the same time?
Yes, there is one function that is both even and odd. For a function to be both, it must satisfy both conditions: f(-x) = f(x) and f(-x) = -f(x). This implies that f(x) must equal -f(x), which means 2f(x) = 0. The only function that satisfies this for all x in its domain is the zero function, f(x) = 0.
7. How are even and odd functions important in higher-level mathematics like calculus?
The concept of even and odd functions is very important for simplifying problems in higher mathematics. A key application is in calculating definite integrals. For example, the integral of an odd function from -a to +a is always zero, which can save significant calculation time. In engineering and physics, these properties are fundamental to Fourier series, which breaks down complex periodic signals into simpler sine (odd) and cosine (even) waves.
8. Is it true that every function must be either even or odd?
No, this is a common misconception. Most functions, such as f(x) = x + 1, are neither even nor odd. However, a powerful concept in mathematics is that any function whose domain is symmetric about the origin can be uniquely expressed as the sum of an even function and an odd function. This property is crucial in fields like signal processing and mathematical analysis.
