Mastering Even and Odd Functions for Competitive Exams

Understanding the properties of even and odd functions is crucial for simplifying complex problems in calculus and algebra, especially in competitive exams like JEE. These properties can significantly reduce the effort required to evaluate integrals, analyze graphs, and solve equations. Let's dive into what defines these functions and how to identify them.

What are Even and Odd Functions?

Functions can be classified based on their symmetry. Even functions exhibit symmetry about the y-axis, while odd functions display symmetry about the origin. This symmetry is mathematically defined by specific relationships between the function's values at $x$ and $-x$ .

Even functions are symmetric about the y-axis.

An even function, when graphed, looks the same on both sides of the y-axis. Mathematically, this means that for every point $(x, y)$ on the graph, the point $(-x, y)$ is also on the graph.

A function $f(x)$ is defined as an even function if $f(-x) = f(x)$ for all $x$ in its domain. This property implies that replacing $x$ with $-x$ in the function's expression does not change the function's value. Common examples include $f(x) = x^2$ , $f(x) = \cos(x)$ , and $f(x) = |x|$ .

Odd functions are symmetric about the origin.

An odd function, when graphed, has rotational symmetry about the origin. If you rotate the graph 180 degrees around the origin, it remains unchanged. Mathematically, this means that for every point $(x, y)$ on the graph, the point $(-x, -y)$ is also on the graph.

A function $f(x)$ is defined as an odd function if $f(-x) = -f(x)$ for all $x$ in its domain. This property means that replacing $x$ with $-x$ in the function's expression results in the negative of the original function's value. Common examples include $f(x) = x^3$ , $f(x) = \sin(x)$ , and $f(x) = x$ .

What is the defining mathematical condition for an even function?

$f(-x) = f(x)$

What is the defining mathematical condition for an odd function?

$f(-x) = -f(x)$

Identifying Even and Odd Functions

To determine if a function is even, odd, or neither, we substitute $-x$ for $x$ in the function's expression and simplify. We then compare the result with the original function $f(x)$ and $-f(x)$ .

Consider the function $f(x) = x^2 - 4$ . To check if it's even or odd, we evaluate $f(-x)$ : $f(-x) = (-x)^2 - 4 = x^2 - 4$ . Since $f(-x) = f(x)$ , the function is even. Its graph is a parabola symmetric about the y-axis.

Now consider $g(x) = x^3 + x$ . Evaluating $g(-x)$ : $g(-x) = (-x)^3 + (-x) = -x^3 - x = -(x^3 + x)$ . Since $g(-x) = -g(x)$ , the function is odd. Its graph has origin symmetry.

For $h(x) = x^2 + x$ , $h(-x) = (-x)^2 + (-x) = x^2 - x$ . This is neither $h(x)$ nor $-h(x)$ , so $h(x)$ is neither even nor odd.

📚

Text-based content

Library pages focus on text content

Property	Even Function	Odd Function
Symmetry	About the y-axis	About the origin
Condition	$f(-x) = f(x)$	$f(-x) = -f(x)$
Example	$f(x) = x^2$	$f(x) = x^3$

Properties and Applications in Exams

Understanding even and odd functions simplifies many calculus operations. For instance, when integrating over a symmetric interval $[-a, a]$ :

The integral of an odd function is zero: $\int_{-a}^{a} f(x) dx = 0$ if $f(x)$ is odd.
The integral of an even function is twice the integral from $0$ to $a$ : $\int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) dx$ if $f(x)$ is even.

These properties can save significant time in solving definite integral problems in competitive exams.

Remember: A function can be even, odd, or neither. It cannot be both even and odd unless it is the zero function ( $f(x) = 0$ ).

Practice identifying these functions and applying their integral properties to build speed and accuracy for your exams.

Learning Resources

Even and Odd Functions - Khan Academy(video)

This video provides a clear explanation of even and odd functions, their definitions, and how to identify them with examples.

Even and Odd Functions - Mathematics LibreTexts(documentation)

A comprehensive text resource covering the definitions, properties, and graphical interpretations of even and odd functions.

Properties of Even and Odd Functions - Brilliant.org(blog)

Explores the properties of even and odd functions, including their behavior under addition, subtraction, multiplication, and division, with interactive examples.

Understanding Even and Odd Functions - Paul's Online Math Notes(documentation)

Detailed notes on even and odd functions, including algebraic tests, graphical interpretations, and common examples.

Even and Odd Functions - YouTube Tutorial by Professor Leonard(video)

A thorough video lecture explaining the concepts of even and odd functions with detailed examples and problem-solving techniques.

Even and Odd Functions - Wikipedia(wikipedia)

Provides a formal definition, properties, and examples of even and odd functions, including their relationship to Fourier series.

Calculus I - Even and Odd Functions - Lamar University(documentation)

Focuses on the application of even and odd function properties in calculus, particularly for definite integrals.

Identifying Even and Odd Functions - Math is Fun(blog)

A beginner-friendly explanation of even and odd functions with simple examples and visual aids.

JEE Mathematics: Even and Odd Functions Practice Problems(blog)

Offers practice problems and solutions related to identifying and applying properties of even and odd functions, relevant for competitive exams.

Symmetry of Functions - Art of Problem Solving(documentation)

Discusses various types of function symmetry, including even and odd functions, with a focus on problem-solving strategies.