Even and Odd Function: Definition, Graph, Properties and Examples

Even and odd functions are special types of functions that exhibit particular symmetrical properties. When discussing real functions — that is, real-valued functions of a real variable — evenness and oddness are typically taken into account. The ideas, however, might be more broadly stated for functions for which there is an additive inverse notion in both the domain and the codomain. As a result, a complex-valued function of a vector variable, a real function, and so on, can all be odd, even, or neither. Comprehending these functions is crucial for numerous mathematical domains, such as algebra, calculus, and Fourier analysis.

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In this article, we will cover the concepts of the even and odd functions. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of one question has been asked on this concept, including one in 2019.

What is Even function?

For a real-valued function $f(x)$, when the output value of $f(-x)$ is the same as $f(x)$, for all values of $x$ in the domain of $f$, the function is said to be an even function. An even function should hold the following equation: $f(-x)=f(x)$, for all values of $x$ in $D(f)$, where $D(f)$ denotes the domain of the function $f$. In other words, we can say that the equation $f(-x)-f(x)=0$ holds for an even function, for all $x$. Let us consider an example, $f(x)=x^2$.
$f(-x)=(-x{)}^2=x^2$ for all values of $x$, as the square of a negative number is the same as the square of the positive value of the number. This implies $f(-x)=f(x)$, for all $x$. Hence, $f(x)=x^2$ is an even function. Similarly, functions like $x^4,x^6,x^8$, etc. are even functions.

If for a function $f(x),f(-x)=f(x)$ then the function is known as an even function. Even functions are symmetric about the $y$-axis.

$y=x^2$ $y=|x|$ $y=cos(x)$

Graphical Representation: The graph of an even function remains unchanged when reflected across the $y$-axis.

What is the Odd function?

When all values of $x$ in the domain of $f$ for a real-valued function $f(x)$ have an output value of $f(-x)$ equal to the negative of $f(x)$, the function is said to be odd. The following equation ought to hold for an odd function: For all values of $x$ in $D(f)$, where $D(f)$ represents the function f's domain, $f(-x)$ equals $-f(x)$. Put otherwise, we can state that for each $x$, the equation $f(-x)+f(x)=0$ applies to an odd function.
Let us consider an example, $f(x)=x^3$.
$f(-x)=(-x{)}^3=-\left(x^3\right)$ for all values of $x$, as the cube of a negative number is the same as the negative of the cube of the positive value of the number. This implies $f(-x)=-$ $f(x)$, for all $x$. Hence, $f(x)=x^3$ is an odd function. Similarly, functions like $x^5,x^7$, $x^9$, etc. are odd functions.

If for a function $f(x),f(-x)=-f(x)$ then the function is known as an odd function. Odd functions are symmetric about the origin.

$y=sin(x)$ $y=x^3$

Graphical Representation: The graph of an odd function remains unchanged when rotated $180$ degrees around the origin.

Properties of Even and Odd Functions

1. Addition and Subtraction:

   • The sum of two even functions is even.
   • The sum of two odd functions is odd.
   • The sum of an even function and an odd function is neither even nor odd (unless one of them is zero).

2. Multiplication:

   • The product of two even functions is even.
   • The product of two odd functions is even.
   • The product of an even function and an odd function is odd.

3. Integration:

   - The integral of an even function over a symmetric interval $[-3$, a $]$ is twicethe integral from 0 to a: ${\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx$
   - The integral of an odd function over a symmetric interval $[-a,a]$ is zero:${\int }_{-a}^{a}f(x)dx=0$

How to Determine if a Function is Even or Odd

1. Algebraic Method:

   • Substitute $-x$ into the function.
   • Simplify the expression.
   • Check if the resulting expression is equal to $f(x)$ (even) or $-f(x)$ (odd).

2. Graphical Method:

   • Plot the function.
   • Check for symmetry about the y-axis (even) or the origin (odd).

NOTE: We can have functions that are neither even nor odd. Eg, $y=x+1$

Summary

Particular symmetrical qualities are exhibited by even and odd functions: odd functions are symmetric about the origin, whereas even functions are symmetric about the y-axis. Comprehending these characteristics is crucial for evaluating functions in algebra, calculus, and other scientific and engineering applications. Checking the algebraic forms of even and odd functions and applying symmetry considerations are necessary when dealing with them.

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Solved Examples Based On the Even and Odd Functions:

Example 1: The graph of the function $y=f(x)$ is symmetrical about the line $x=2$, then
1) $f(x)=f(-x)$
2) $f(2+x)=f(2-x)$
3) $f(x+2)=f(x-2)$
4) $f(x)=-f(-x)$

Solution:

Since a graph is symmetric about the $y$-axis means $x=0$ then it is even function and $f(-x)=f(x)$
$\therefore \mathrm{f}(0-\mathrm{x})=\mathrm{f}(0+\mathrm{x})(\mathrm{b}<\mathrm{z}$ it is symmetric about $\mathrm{v}=0)$
But in question, it is symmetric about $x=2$ then $f(2-x)=f(2+x)$

Hence, the answer is option 2 .

Example 2: Which of the following functions is an even function?

1) $x^6+3x^3-5$
2) $x^6+3x^4-5$
3) $x^6+3x^3-5x$
4) $x^6+3x^4-5x$

Solution:
|f $f(x)=x^6+3x^4-5$

$f(-x)=x^6+3x^4-5=f(x)$
Hence, it is an even function

Hence, the answer is option 2.

Example 3: Which of the following functions is an even function?

1) $\sin x$
2) $\cos x$
3) $\tan x$
4) $\csc x$

Solution:
Even Function -

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

Symmetric about ${\gamma }_{\text{-axis}}$
If $f(x)=\cos x$
then $f(-x)=\cos (-x)=\cos x$
Hence, the answer is option 2 .

Example 4: A function $f$ from the set of natural numbers to integers defined by
$f(n)=\{\begin{array}{l}\frac{n-1}{2},\text{when}n\text{is odd}\\ -\frac{n}{2},\text{when}n\text{is even is}\end{array}$

1) onto but not one­-one

2) one­-one and onto both

3) neither one-­one nor onto

4) one­-one but not onto.

Solution:

$f(x)=\{\begin{array}{ll}\frac{n-1}{2}& \text{when}n\text{is odd}\\ -\frac{n}{2}& \text{when}n\text{is even}\end{array}$

$f(1)=0$

$f(2)=-1$

$f(3)=1$

$f(4)=2$

$f(5)=2$

Clearly, this pattern will cover all the integers, hence onto

Also, it is a one-one function

Hence, the answer is option 2

Example 5: The function $f(x)=\log (x+\sqrt{x^2+1})$
1) an odd function
2) a periodic function
3) neither an even nor an odd function
4) an even function.

Solution:
As we learned in
Odd Function -

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

- wherein

Symmetric about origin

$\begin{array}{rl}& f(x)=\log (x+\sqrt{1+x^2})\\ & f(-x)=\log (-x+\sqrt{1+x^2})\\ & f(x)+f(-x)=\log (\sqrt{1+x^2}+x)+\log (\sqrt{1+x^2}-x)\\ & =\log (1+x^2-x^2)=\log 1=0\end{array}$
It is an odd function.

Hence, the answer is the option 1.