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How to Determine if a Function is Even, Odd or Neither | Step by Step

 

I have prepared eight (8) worked examples to illustrate the procedure or steps on how to figure out if a given function is even, odd, or neither. The math involved in the calculation is easy as long as you are careful in every step of your solution.

To get into the "heart" of this topic, study the illustration below.

 

Given f(x), what do I get if I plug in -x? even function: f(-x)=f(x ), odd function:f(-x)=-f(x)

 

Let us talk about each case...

 

Case 1: Even Function

Given some "starting" function f(x):

If we evaluate or substitute into f(x) and get the original or "starting" function again, that implies f(x) is an even function.

Got the original function. This is even since f(-x)=f(x).
Case 2: Odd Function

Given some "starting" function f(x):

If we evaluate or substitute -x into f(x) and get the negative or opposite of the "starting" function, that implies f(x) is an odd function.

odd function since f(-x)=-f(x)
Case 3: Neither

Given some "starting" function f(x):

If we evaluate or substitute -x into f(x) and we don't obtain either Case 1 or Case 2, that implies f(x) is neither even nor odd.  In other words, it does not fall under the classification of being even or odd.

Since f(-x)≠f(x) , and f(-x)≠-f(x) then this is a "neither" function.

 


Directions: Determine algebraically whether given function is even, odd, or neither.

 

Example 1 f(x)=2x^2-3 See solution
Example 2 f(x)=-3x^3+2x See solution
Example 3 f(x)=3x^6-5x^4+6x^2-1 See solution
Example 4 f(x)=-x^7+8x^5-x^3+6x See solution
Example 5 f(x)=10x^3-4x^2+3x-8 See solution
Example 6 g(x)=-7x^7-x^3+5x See Solution
Example 7 h(x)=x^3-x^2-1 See Solution
Example 8 k(x)=-5x^8+7x^4-3x^2+5 See Solution

 


Example 1: Determine algebraically whether the given function is even, odd, or neither: f(x)=2x^2-3

I start with the given function f(x)=2x^2-3, plug in the value − x and then simplify. What do I get? Let us work it out algebraically.

f(-x)=2(-x)^2-3 = 2x^2-3. I got back the original function. Therefore this is an even function.

 

Since f(-x)=f(x), it means f(x) is an even function!

The graph of an even function is symmetric with respect to the y-axis, or along the vertical line x=0. Observe that the graph of the function is cut evenly at the y-axis and each half is an exact mirror of the another. Another way of describing it is that each half of the function is a reflection across the y-axis.

See the animated illustration.

animated gif of  f(x)=2x^2-3  showing that it is symmetric along y-axis or at x=0

 


Example 2: Determine algebraically whether the given function is even, odd, or neither: f(x)=-3x^3+2x

I will substitute − x into the function f(x)=-3x^3+2x, and then simplify.

 

f(-x)=-3(-x)^3+2(-x)=-(-3x^3+2x) therefore this is an odd function.

 

tips Tips

  • If ever you arrive at a different function after evaluating − x into the given f(x), immediately try to factor out −1 from it, and observe if the original function shows up. If it does, then we have an odd function.

 

  • The effect of factoring out −1 results to the switching of the signs of the terms inside the parenthesis. This is a key step to identify an odd function.

 

Now, since f(-x)=-f(x), it implies that the original function f(x)is an odd function!

The graph of an odd function has rotational symmetry about the origin, or at the point (0,0). That means we cut its graph along the y-axis and then reflect its even half in the x-axis first followed by the reflection in the y-axis.

See the animated illustration.

graph of the odd function f(x)= -3x^3+2x which is symmetrical with respect to the origin.

 

 


Example 3: Determine algebraically whether if the function is even, odd, or neither: f(x)=3x^6-5x^4+6x^2-1

Here I observed that the exponents of variable x are all even numbers; namely 6,4, and 2. As for the constant term, I must add that it can also be expressed as -1=-1x^0 which has an even power of zero.

This characteristic of a function containing only of even powers can likely result to an even function. However, we must show it algebraically, so here it goes.

Evaluating − x into f(x), I have the following algebraic steps...

This is an even function because f(-x)=3(-x)^6-5(-x)^4+6(-x)^2-1=3x^6-5x^4+6x^2-1=f(x)

It is clearly an even function!

 


Example 4: Determine whether the given function is even, odd, or neither:

In contrast to example 3 where the function has even powers, this one has odd powers which are 7, 5, 3 and 1. By now, I hope you're already seeing the pattern. This is more likely an odd function, right?

Substituting − x into the given f(x), and simplifying...

This is an odd function for the reason of  f(-x) = -(-x^7+8x^5-x^3+6x)=-f(x)

After factoring out −1, the polynomial inside the parenthesis equals the starting function. It shows that this is an odd function!

 


Example 5: Determine whether the given function is even, odd, or neither: f(x)=10x^3-4x^2+3x-8

This time I will show you an example of a function that is neither even nor odd. Are you ready?

  • First, check if it is even. Do we have the case f(-x)=f(x) ?
Not an even function since f(-x)=-10x^3-4x^2-3x-8 is not equal to f(-x)=f(x)=10x^3-4x^2+3x-8.

Definitely not an even function since f(-x)≠ f(x).

  • Secondly, check if it is odd by showing f(-x)= - f(x).
Not an odd function either since f(-x)≠-f(x)

Even after factoring out −1, I still don't get the original function.

This is not an odd function since f(-x)≠-f(x).

 

  • Conclusion: Since we reached the case where f(-x)≠f(x) and f(-x)≠-f(x) , this function is neither even nor odd!

 


Example 6: Determine whether the given function is even, odd, or neither: g(x)=-7x^7-x^3+5x

 

g(x) is odd function since g(-x) = -(-7x-x^3+5x)=-g(x)

 

Therefore, function g(x) is an odd function!

 


Example 7: Determine whether the given function is even, odd, or neither: h(x)=x^3-x^2-1

shows the solution why h(x)=x^3-x^2-1 is neither since h(-x)≠h(x), and h(-x)≠-h(x).

Therefore, function h(x) is neither !

 

 


Example 8: Determine whether the given function is even, odd, or neither: k(x)=-5x^8+7x^4-3x^2+5

shows the solution why k(x)=-5x^8+7x^4-3x^2+5 is an even function because k(-x)=k(x)

 

Therefore, function k(x) is an even function!

 


 

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About This Topic: Even, Odd, or Neither Functions

 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

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