Even Function - Definition, Properties, Graph, Examples - Cuemath

Even Function

A function is an even function if f of x is equal to f of −x for all the values of x. This means that the function is the same for the positive x-axis and the negative x-axis, or graphically, symmetric about the y-axis. An example of an even function are the trigonometric even function, secant function, etc. Let us explore more about the even function in detail along with its graphical representation and properties.

1.	What is an Even Function?
2.	Graphical Representation of an Even Function
3.	Properties of an Even Function
4.	FAQs on Even Function

What is an Even Function?

Let us first understand the meaning of even functions, algebraically. A function is even if f(x) = f(- x) for all values of x. Now, let us see what this means. For an even function f(x), if we plug in -x in place of x, then the value of f(-x) is equal to the value of f(x). Similarly, functions like \(x^4, x^6, x^8, x^{10}\), etc. are even functions.

Interestingly, the above functions have even powers. Observe the graph below y = x2, an even function graph.

Example of an Even Function

Consider a trigonometric function (f(x) = cos x. Determine the value of f(-x) and identify if it is an even function or not.

Solution: f(-x) = cos (-x) = cos x = f(x)

cos (-x) = cos x for all values of x

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

Graphical Representation of an Even Function

Let us now see how an even function behaves graphically. The above graph of an even function is symmetric with respect to the y-axis. In other words, the graph of an even function remains the same after reflection about the y-axis.

Here are a few examples of even functions, observe the symmetry about the y-axis.

Let us see a plot for f(x) = cos x

Properties of an Even Function

After understanding the even function meaning, we are going to explore its properties. A few major properties of an even function are listed below.

• The sum of two even functions is even.
• The difference between the two even functions is even.
• The product of two even functions is even.
• The quotient of the division of two even functions is even.
• The composition of two even functions is even.
• The composition of an even and odd function is even.

☛Articles on Even Function

Given below is the list of topics that are closely connected to the even function. These topics will also give you a glimpse of how such concepts are covered in Cuemath.

• Exponential Function
• Polynomial Functions
• Quadratic Functions
• Linear Functions
• Constant Functions