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Math Lesson 16.7.4 - Definition of Odd Functions

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Welcome to our Math lesson on Definition of Odd Functions, this is the fourth lesson of our suite of math lessons covering the topic of Even and Odd Functions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Definition of Odd Functions

Now that you know what an even function is, it is easier to understand the definition of odd functions without long introductions. Thus, by definition, an odd function f(x) is a function for which f(-x) is always equal to -f(x).

In other words, If f(-x) = - f(x) for any x-value, then the function f(x) is odd.

Thus, the function f(x) = 2x is odd because, for example, for x = 3, we have f(-x) = f(-3) = -2 · (-3) = -6 and for x = 3 we have -f(x) = -f(3) = -2 · 3 = -6.

Again, we can prove the oddness of a function in two ways: by substitution of some values and analytically.

Example 4

Check whether the functions below are odd by substituting some values in them.

  1. f(x) = 2x - 3
  2. g(x) = x3
  3. h(x) = √x
  4. i(x) = 3x - x3

Solution 4

  1. Let's choose some small values to make the calculations easier. Thus, For x = 3,
    f(-3) = 2 ∙ (-3) - 3
    = -6 - 3
    = -9
    and
    -f(3) = -(2 ∙ 3 - 3)
    = -(6 - 3)
    = -3
    Since for this value (x = 3) we obtained two different values for f(-x) and -f(x), it is not necessary to continue further, as now it is clear that the function f(x) is not odd.
  2. For x = 1, we have
    g(-1) = (-1)3
    = -1
    and
    -g(1) = -13
    = -1
    For x = 2, we have
    g(-2) = (-2)3
    = -8
    and
    -g(2) = -23
    = -8
    For x = 3, we have
    g(-3) = (-3)3
    = -27
    and
    -g(3) = -33
    = -27
    and so on. Therefore, since g(-x) = -g(x) for any value, the function g(x) is odd.
  3. In the function h(x) we have for x = 1, h(-x) is not defined because √(-1) is not defined in R while
    -h(x) = -√1
    = -1
    Therefore, since we obtained two different outputs, the function h(x) is not odd.
  4. Let's check some values in the function i(x). Thus, for x = 1, we have
    i(-x) = i(-1) = 3 ∙ (-1) - (-1)3
    = -3 + 1
    = -2
    and
    -i(x) = -i(1) = -(3 ∙ 1 - 13)
    = -(3 - 1)
    = -2
    Likewise, for x = 2 we have
    i(-x) = i(-2) = 3 ∙ (-2) - (-2)3
    = -6 + 8
    = 2
    and
    -i(x) = -i(2) = -(3 ∙ 2 - 23)
    = -(6 - 8)
    = -(-2)
    = 2

You have reached the end of Math lesson 16.7.4 Definition of Odd Functions. There are 10 lessons in this physics tutorial covering Even and Odd Functions, you can access all the lessons from this tutorial below.

More Even and Odd Functions Lessons and Learning Resources

Functions Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
16.7Even and Odd Functions
Lesson IDMath Lesson TitleLessonVideo
Lesson
16.7.1Definition of Even Functions
16.7.2How to Prove the Evenness of a Function Analytically
16.7.3The Graph of Even Functions
16.7.4Definition of Odd Functions
16.7.5Proving the Oddness of a Function Analytically
16.7.6The Graph of Odd Functions
16.7.7Conclusions about the Evenness and Oddness of a Function
16.7.8What If a Function is Neither Even Nor Odd?
16.7.9Properties of Even Functions
16.7.10Properties of Odd Functions

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