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# Even Function

A function can be defined as even, odd or neither in different ways, either algebraically or graphically. A function is called an even function if its graph is unchanged under reflection in the y-axis. Suppose f(x) is a function such that it is said to be an even function if f(-x) is equal to f(x). In this article, you will learn the mathematical definition of even function, formula, graph, properties, along with examples.

Learn: What is a function?

## Even Function Definition

Consider a function f(x), where x is a real number. Here, the function f(x) is called an even function when we substitute -x in the place of x and get the expression the same as the original function. That means, the function f(x) is called an even function if f(-x) = x for all real values of x.

 Even function: f(-x) = f(x) Odd function: f(-x) = -f(x)

## Even Function Graph

The graph of the even function is symmetric with respect to the y-axis. That means the graph of the even function remains the same when the y-axis acts like a mirror. The figure given below, shows the graph of an even function. Here, we can see that the curve in the graph is symmetric about the y-axis. That means, even if we flip the graph vertically, we can see the same curved shape.

### Even Function Examples

Some of the examples of even functions include the following.

• cos x since cos(-x) = cos x
• x2, x4, x6, x8,…, i.e. xn is an even function when n is an even integer
• |x|
• cos2x
• sin2x
• cos nx

### Even Function Formula

The formula of an even function is simply the expression that helps to identify whether a function is even.

Function f(x) = even if f(-x) = f(x)

Using this, we can check whether the given function is even or odd.

This expression is used to check for the functions algebraically.

### Even Function Properties

The important properties of even functions are listed below:

• For any function f(x), f(x) + f(−x) is an even function.
• The sum or difference of two even functions is even.
• The multiple of an even function is again an even function.
• The product or division of two even functions is even.

For example, x2 cos(x) is an even function where x2 and cos x are even.

In the case of division, the quotient of two even functions is even.

• The derivative of an odd function is an even function.
• The composition of two even functions and the composition of an even and odd function is even. This can be represented as:

f(g(x)) is an even function

Here, f(x), g(x) are even

or

f(x) is even, g(x) is odd and vice versa

Thus, f(g(−x)) = f(−g(x)) = f(g(x))

### Even Function and Odd Function

Even function and odd function can be defined as given below:

 Even function Odd function f(-x) = f(x) f(-x) = -f(x) The graph of an even function is symmetric with respect to the y-axis. The graph of an odd function is symmetric with respect to the origin. Here, the y-axis acts like a mirror. That means if we flip vertically, the graph looks the same. Here, if we spin the graph upside down about the origin, it looks the same. Examples: x2, x4,… cos x Examples: x, x3, x5,.. sin x

Example of Even Function and Odd Function

The graph below shows both even and odd functions.

### Here, sin x is the odd function, whereas cos x is the even function. Solved Problems

Question 1:

Identify the even function and odd function from the following graph: Solution:

From the given graph,

Even function is y = x2

Odd function is y = x7

Question 2:

Check which of the following functions are even.

(i) f(x) = x3 + 2x

(ii) f(x) = x4 + 7

(iii) f(x) = x2 + 3x – 5

Solution:

(i) f(x) = x3 + 2x

Let us find f(-x).

f(-x) = (-x)3 + 2(-x) = -x3 – 2x = -1(x3 + 2x) = -f(x)

Thus, the function f(x) = x3 + 2x is not an even function, but an odd function.

(ii) f(x) = x4 + 7

f(-x) = (-x)4 + 7 = x4 + 7 = f(x)

Therefore, f(x) = x4 + 7 is an even function.

(iii) f(x) = x2 + 3x – 5

f(-x) = (-x)2 + 3(-x) – 5 = x2 – 3x – 5

Here, f(-x) ≠ f(x) and f(-x) ≠ -f(x)

Thus, the function f(x) = x2 + 3x – 5 is neither even nor odd.