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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 xaxis and the negative xaxis, or graphically, symmetric about the yaxis. 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 = x^{2}, 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 yaxis. In other words, the graph of an even function remains the same after reflection about the yaxis.
Here are a few examples of even functions, observe the symmetry about the yaxis.
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.
Even Function Examples

Example 1: Sam wants to determine algebraically if the function f(x) = 4x^{4} − 7x^{2} is an even function or not.
Solution: Substitute x in place of x in f(x) = 4x^{4}  7x^{2}.
f(−x) = 4(−x)^{4} −7(−x)^{2}
= 4x^{4} − 7x^{2}
= f(x)Since f(−x) = f(x), the function f(x) is an even function.

Example 2: Consider the function f(x) = x^{2}. Determine the value of f(−x). Identify if it is an even function or not.
Solution: f(−x) = (−x)^{2} = x^{2} = f(x)
Therefore, f(x) = x^{2} is an even function.
We can verify by taking a particular value of x.
For x = 2, the value of f(x) is given by:
f(2) = 2^{2} = 4
The value of f(−x) is given by:
f(−2) = (−2)^{2} = 4 = f(2)

Example 3: Determine if the function f(x) = 6x^{4} − x^{12} is even function or not.
Solution: Substitute −x in place of x in f(x) = 6x^{4} − x^{12}.
f(−x) = 6(−x)^{4} − (−x)^{12}
= 6x^{4} − x^{12}
= f(x)
Since f(−x) = f(x), the function f(x) is an even function.
f(x) = 6x^{4} − x^{12} is an even function.
FAQs on Even Function
What are Even Functions in Calculus?
Even functions are those functions in calculus which are the same for +ve xaxis and ve xaxis, or graphically, symmetric about the yaxis. It is represented as f(x) = f(x) for all x. Few examples of even functions are x^{4}, cos x, y = x^{2}, etc.
What is the Even Function Equation?
The even function equation mathematically expressed as f(x) = f(x) for all x.
How to Determine if a Function is an Even Function or Not?
If a function satisfies f(−x) = f(x) for all x it is said to be an even function. That means it is the same for +ve xaxis and ve xaxis, or graphically, symmetric about the yaxis.
 If the value of f(−x) is the same as the value of f(x) for every value of x, the function is even.
 If the value of f(−x) is NOT the same as the value of f(x) for any value of x, the function is not even.
 If a function has an even power, the function need not necessarily be an even function.
Is Cos x an Even Function?
The even function equation mathematically expressed as f(−x) = f(x) for all x. On substituting the value we have cos(−x) = cosx. Therefore, cos x is an even function.
How Do You Identify if a Plotted Graph is of an Even Function?
There are certain rules to identify whether the plotted graph is of an even function or not. The rules are listed below.
 If a graph is symmetrical about the y axis, the function is even.
 If a graph is symmetrical about the origin, the function is odd.
 If a graph is not symmetrical about the yaxis or the origin, the function is neither even, nor odd.
Are Constants Even Function?
A constant function f(x) = k is an even function because f(−x) = k = f(x).
Write Two Major Properties of an Even Function.
There are various properties that define an even function. The two major properties are:
 When we subtract two even functions the difference is even.
 When we multiply two even functions the product is even.
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