Even Function

Even Function

The teacher asked the students to observe the following shapes and find the kind of symmetry seen in them.

Luke said that the shapes looked to be vertically symmetrical.

So, let us study the two types of functions, on the basis of their symmetry, to learn more about the concept.

Even function symmetric about the Y-axis

Lesson Plan


Classification of Functions

Functions can be even, odd, both, or neither of them.

Let us explore some even and odd function examples in this page.

Even Function

A function is even if \(f(x) = f( - x)\)  for all \(x\)

This means that the function is the same for \(\text{+ve } x\text{-axis}\) and \(\text{-ve } x\text{-axis,}\) or graphically, symmetric about the \(y\text{-axis}\).

For example, \(f(x) = {x^2}\)

even function graph showing y=x^2

Odd Function

A function is odd if \( -f(x)=  f(-x)\)  for all \(x\).

The graph of an odd function is symmetrical about the origin.

For example, \(f(x) = {x^3}\) is odd.

odd function showing y=x^3

Neither Even, Nor Odd

A function \( f(x)\)  which is \(f(x) \neq  f( - x)\) and  \(-f(x) \neq  f(-x)\)  for all \(x\)   is neither an even function, nor an odd function.

For example, \(f(x)=2x^{5}+3x^{2}+1\)

neither even nor odd function graph

We can see that the graph is not symmetrical about the origin or the y-axis.

Both Odd and Even

A function is both odd and even, if  \( -f(x)=  f(-x)\)  and  \(f(x) = f( - x)\)  for all \(x\) .

The only function which is both odd and even is \(f(x) = 0 \)

Odd and even function

The graph is symmetrical about the origin and y-axis.


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 \(x\).

This means that the function is the same for \(\text{+ve } x\text{-axis}\) and \(\text{-ve } x\text{-axis,}\) or graphically, symmetric about the \(y\text{-axis}\).

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 \(f(x\)).

Formula of Even Function

\(f(-x)=f(x)\) for all \(x\)

Examples of Even Functions

1. Consider the function \(f(x)=x^2\)

Determine the value of \(f(-x)\).

\(f(-x)=(-x)^2=x^2=f(x)\)

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:

\(f(-2)=(-2)^2=4=f(2)\)

Similarly, functions like \(x^4, x^6, x^8, x^{10}\), etc. are even functions.

Interestingly, the above functions have even powers.

2. Consider a trigonometric function \(f(x)=\cos x\)

Determine the value of \(f(-x)\).

\(f(-x)=\cos (-x)=\cos x=f(x)\)

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

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

Later, in this section, we will explore the properties of even functions and their graphical representation.

 
tips and tricks
Tips and Tricks
  1. If the value of \(f(-x)\) is same as the value of \(f(x)\) for every value of \(x\), the function is even.
  2. 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.
  3. If a function has an even power, the function need not be an even function.

Graphical Representation of an Even Function

Let us now see how an even function behaves graphically.

The 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.

Even and Odd Functions

Example

\(y=x^2\)

parabola graph

\(f(x)=\cos x\)

cos x graph

 
important notes to remember
Important Notes
  1. An even function has a reflection about the y-axis.
  2. An even function follows a rule: \(f(-x)=f(x)\) for all values of \(x\).
  3. The function \(x^n\) is even for all even values of \(n\).

Properties of an Even Function

After understanding the even function meaning, we are going to explore its properties.

  1. The sum of two even functions is even.
  2. The difference of the two even functions is even.
  3. The product of two even functions is even.
  4. The quotient of two even functions is even.
  5. The composition of two even functions is even.
  6. The composition of an even and odd function is even.

Solved 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 \(f(x)=4x^4-7x^2\).

\(\begin{align}f(-x)&=4(-x)^4-7(-x)^2\\&=4x^4-7x^2\\&=f(x) \end{align}\)

Since \(f(-x)=f(x)\), the function \(f(x)\) is an even function.

\(\therefore f(x)=4x^4-7x^2\) is an even function. 

Example 2

 

 

Determine if the function \(f(x)=6x^4-x^{12}\) is even or not.

Plot its graph and state your observations.

Solution

Substitute \(-x\) in place of \(x\) in \(f(x)=6x^4-x^{12}\).

\(\begin{align}f(-x)&=6(-x)^4-(-x)^{12}\\&=6x^4-x^{12}\\&=f(x) \end{align}\)

Since \(f(-x)=f(x)\), the function \(f(x)\) is an even function.

graph of an even function

From the above graph, we observe the following

  •  The graph is symmetric about the y-axis.
  • For every point \((-x,y)\) on the graph, there's a point \((x,y)\) on the graph.
\(\therefore f(x)=6x^4-x^{12}\) is an even function.

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the 'Check Answer' button to see the result.

 
 
 
 
 

Let's Summarize

We hope you enjoyed learning about Even Function with the solved examples and practice questions. Now you will be able to easily solve problems related to even and odd functions, even function equations, and even function graphs.

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Frequently Asked Questions (FAQs)

1. Is \(\cos x\) an even function?

\(\cos (-x)=\cos x\)

Therefore, \(\cos x\) is an even function.

2. Are constants even or odd functions?

A constant function \(f(x)=k\) is an even function because \(f(-x)=k=f(x)\).

3. How do you tell if a graph is odd, even, or neither of them?

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 y-axis or the origin, the function is neither even, nor odd.

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