Return to the Purplemath home page

 The Purplemath Forums
Helping students gain understanding
and self-confidence in algebra


powered by FreeFind

 

Return to the Lessons Index  | Do the Lessons in Order  |  Get "Purplemath on CD" for offline use  |  Print-friendly page

Function Notation: Even and Odd (page 3 of 3)

Sections: Introduction & Evaluating at a number, Evaluating at a variable, Even and odd functions


You may be asked to "determine algebraically" whether a function is even or odd. To do this, you take the function and plug x in for x, and then simplify. If you end up with the exact same function that you started with (that is, if f(x) = f(x), so all of the signs are the same), then the function is even. If you end up with the exact opposite of what you started with (that is, if f(x) = f(x), so all of the "plus" signs become "minus" signs, and vice versa), then the function is odd. In all other cases, the function is "neither even nor odd".

  • Determine algebraically whether f(x) = 3x2 + 4 is even, odd, or neither.
     

    If I graph this, I will see that this is "symmetric about the y-axis"; in other words, whatever the graph is doing on one side of the y-axis is mirrored on the other side:

     

    This mirroring about the axis is a hallmark of even functions.

    Note also that all the exponents are even (the exponent on the constant term being zero: 4x0 = 4 1 = 4).

    But the question asks me to make the determination algebraically, which means that I need to do it with algebra, not with graphs.

     

    graph of –3x^2 + 4

    So I'll plug x in for x, and simplify:

      f(x) = 3(x)2 + 4
               = 3(x2) + 4
               = 3x2 + 4

    My final expression is the same thing I'd started with, which means that  f(x) is even.

  • Determine algebraically whether f(x) = 2x3 4x is even, odd, or neither.
     

    If I graph this, I will see that it is "symmetric about the origin"; that is, if I start at a point on the graph on one side of the y-axis, and draw a line from that point through the origin and extending the same length on the other side of the y-axis, I will get to another point on the graph.

    This symmetry is a hallmark of odd functions.

        Copyright Elizabeth Stapel 1999-2011 All Rights Reserved

    Note also that all the exponents are odd (since the second term is 4x = 4x1). This is a useful clue. I should expect this function to be odd.

     

    graph of 2x^3 – 4x

    But the question asks me to make the determination algebraically, so I'll plug x in for x, and simplify:

     

    ADVERTISEMENT

     

      f(x) = 2(x)3 4(x)
               = 2(x3) + 4x
               = 2x3 + 4x

    My final expression is the exact opposite of what I started with, by which I mean that the sign on each term has been changed to its opposite, just as if I'd multiplied through by 1:

      f(x) = 1[f(x)]
               = [2x3 4x]
               = 2x3 + 4x

    This means that  f(x) is odd.

  • Determine algebraically whether f(x) = 2x3 3x2 4x + 4 is even, odd, or neither.
     

    This function is the sum of the previous two functions.

    Note that its graph does not have the symmetry of either of the previous ones, nor are all its exponents either even or odd.

     

    I would expect this function to be neither even nor odd.

     
     

    I"ll plug x in for x, and simplify:

     

    graph of 2x^3 – 3x^2 – 4x + 4

      f(x) = 2(x)3 3(x)2 4(x) + 4
               = 2(x3) 3(x2)  + 4x + 4
               = 2x3 3x2 + 4x + 4

    This is neither the same thing I started with (namely, 2x3 3x2 4x + 4) nor the exact opposite of what I started with (namely, 2x3 + 3x2 + 4x 4). This means that

      f(x) is neither even nor odd.

You may find it helpful, when answering this "even or odd" type of question, to write down f(x) and f(x) explicitly, and then compare them to whatever you get for  f(x). This can help you make a confident determination of the correct answer.


You can use the Mathway widget below to practice figuring out if a function is even, odd, or neither. Try the entered exercise, or type in your own exercise. Then click "Answer" to compare your answer to Mathway's. (Or skip the widget and return to the index.)

(Clicking on "View Steps" on the widget's answer screen will take you to the Mathway site, where you can register for a free seven-day trial of the software.)

<< Previous  Top  |  1 | 2 | 3  |  Return to Index

Cite this article as:

Stapel, Elizabeth. "Function Notation: Even and Odd." Purplemath. Available from
    http://www.purplemath.com/modules/fcnnot3.htm. Accessed
 

 



Purplemath:
  Linking to this site
  Printing pages
  School licensing


Reviews of
Internet Sites:
   Free Help
   Practice
   Et Cetera

The "Homework
   Guidelines"

Study Skills Survey

Tutoring from Purplemath
Find a local math tutor


This lesson may be printed out for your personal use.

Content copyright protected by Copyscape website plagiarism search

  Copyright 1999-2012  Elizabeth Stapel   |   About   |   Terms of Use

 

 Feedback   |   Error?