Even Function

Even function is defined as a function that follows the relation f(-x) equals f(x), where x is any real number. Even functions have the same range for positive and negative domain variables. Due to this, the graph of even functions is always symmetric about the Y-axis in cartesian coordinates.

In this article, we will learn about even functions, their examples, properties, graphical representation of even functions, some solved examples, and practice questions related to even functions.

What is an Even Function?

Even Function is a function that has the same output for a corresponding input with different signs, i.e. positive or negative. It can be said that the output of an even function depends only upon the absolute value of the input variable.

Owing to this property, the graph of even functions is symmetric about the Y-axis in cartesian coordinates.

Even Function Definition

Even function is defined as a function which follows the relation that f(-x) equals to f(x), for every real number x in the domain of the function.

Even functions yield same expression if we substitute -x in place of x, i.e. f(-x) = f(x). Some examples of even functions are polynomials which include only even exponents of variables, trigonometric functions such as cos x, sec x, etc.

Even Function Formula

A function is said to be an even function if and only if it satisfies the given formula:

f(-x) = f(x)

For all x ∈ 𝚁

A function which follows the above equality is an even function, otherwise it is not.

Examples of Even Functions

Some examples of even functions are listed as follows:

• cos x
• sec x
• x²ⁿ , where n is a natural number
• Modulus Function, i.e. |x|
• sin²x
• |x| + cos nx, where n is a natural number

Note: To check for an even function, substitute -x in place of x in the expression of f(x), if the obtained expression is same as f(x), the function is an even function otherwise not.

Example: Check whether f(x) = cos x is an even function or not.

Solution:

Given f(x) = cosx

To check f(x) = cos x is an even function

Put x = -x in the given function, we get

f(-x) = cos (-x)

f(-x) = cos x [As cos (-x) = cos x]

Hence, we have f(-x) = f(x)

Therefore, the given function f(x) = cos x is an even function.

Graphical Representation of an Even Function

In the graphical representation of an even function, the curve is always symmetric about Y-axis. In other words, the value of f(x) remains constant irrespective of the sign on x (positive or negative). Few examples of graph of even functions are given below:

Properties of an Even Function

Even Functions holds the following properties:

• Addition or subtraction of two even functions is an even function.

• Product of two even functions is also an even function.

• The plot of even functions exhibit symmetry about Y-axis.

• Average value of even functions over a symmetric interval around the origin is the functional value at x=0, i.e. f(0).

• For any even function, f(-x) = f(x).

• Composition of an even and odd function is also an even function.

• Composition of two even functions is an even function.

Even Function and Odd Function Difference

The difference between even and odd functions is illustrated as follow:

Even Function	Odd Function
Even function is the one which doesn’t have any change in output if sign of input is changed.	Odd function is the one in which sign of the output is changed if sign of input is changed but output value remains same.
An even function follows the below given equality: f(-x) = f(x).	An odd function follows the below given equality: f(-x) = -f(x)
The plot of an even function is symmetrical about Y-axis.	The plot of an odd function has a rotational symmetry about the origin.
Examples of even functions include: x2 , cos x, x4, etc.	Examples of odd functions include: x3 , sin x, x, etc.

Read More,

• Relation and Function
• Domain and Range of a Function
• Types of Function

Solved Examples on Even Function

Example 1: Check whether the function f(x) = x²+ 2x is even or not?

Solution:

For an even function, f(-x) = f(x)

We have, f(x) = x² + 2x

Now, f(-x) = (-x)² + 2(-x) = x² – 2x

We see that, f(-x) ≠ f(x)

Thus, f(x) is not an even function.

Example 2: State whether f(x) = x² + cos(x) is an even function or not?

Solution:

We have, f(x) = x² + cos(x)

We know that, x² and cos(x) are even functions. Also, addition of two even functions is even.

So, the given function f(x) = x² + cos(x) is an even function.

Example 3: Consider the function, f(x) = e^2x. State whether f(x) is even or not?

Solution:

We have, f(x) = e^2x

On putting -x in place of x, we get,

⇒ f(-x) = e^2(-x) = e^-2x = 1/e^2x

⇒ We have, e^2x ≠ 1/e^2x

Thus, f(-x) ≠ f(x)

Hence, f(x) is not an even function.

Example 4: Determine whether the function, f(x) = x⁴ tan²(x) is even or not?

Solution:

Here, we have product of two functions in f(x), i.e. x⁴ and tan²(x).

Let, g(x) = x⁴ and h(x) = tan²(x)

Substituting -x in g(x), we get,

g(-x) = (-x)⁴ = x⁴ = g(x)

Thus, g(x) is an even function.

Similarly for h(x), we have,

h(-x) = tan²(-x) = (-tan(x))² = tan²x = h(x)

Hence, h(x) is also an even function.

As, product of two even functions is an even function, we get f(x) is also an even function.

Practice Questions on Even Functions

Question 1: Check whether following functions are even or not:

• 2x⁴+x²+5
• cos(3x)
• e^x
• sin²x + tan²x
• sin(2x)

Question 2: Prove that following functions are even functions:

• 4x² + cos(x)
• 1 – 2sin²(x)

Question 3: Determine whether following function is even or not: f(x) = x⁴ + 2x + 1.

Question 4: State whether following function is even or not: g(x) = cos(x³).

Question 5: Prove that following function is even: 4cos³x – 3cos(x).

Even Function: Frequently Asked Questions

How to check a function is even or not?

For an even function, f(-x) = f(x) equality holds true for every real number x for which f(x) is defined.

Give some examples of even functions.

Examples of even functions are x², cos(x), sec(x), |x|, etc.

Are all polynomial functions even?

No, polynomial functions which involve only even powers of the independent variable are even functions. Those involving odd powers wouldn’t follow the property f(-x) = f(x).

Can a function be even as well as odd?

Yes, the zero function is both even and odd because for all values of x, f(-x) = -f(x) = f(x) = 0.

Is Cos x an even function?

Yes, cos x is an even function as cos(-x) = cos x.