Math Lesson 16.7.10 - Properties of Odd Functions

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Welcome to our Math lesson on Properties of Odd Functions, this is the tenth 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.

Properties of Odd Functions

The properties of odd functions are similar to those of even functions. Thus, if two odd functions f(x) and g(x) are given, the following properties are true:

1. The sum of two odd functions is always an odd function.In symbols,
   f(x) = odd & g(x) = odd ⟹f(x) + g(x) = odd
   For example, if f(x) = 2x and g(x) = x³, then
   f(x) + g(x) = 2x + x³
   which is an odd function as well, as it contains only odd-degree monomial terms.
2. The difference of two odd functions is always an odd function.In symbols,
   f(x) = odd & g(x) = odd ⟹ f(x) - g(x) = odd
   For example, if f(x) = 3x³ and g(x) = 2x, then
   f(x) - g(x) = 3x³ - 2x
   which is an odd function as well, as it contains only odd-degree monomial terms.
3. The product of two odd functions is always an even function. In symbols,
   f(x) = odd & g(x) = odd ⟹ f(x) ∙ g(x) = even
   For example, if f(x) = x and g(x) = 3x⁵, then
   f(x) ∙ g(x) = x ∙ 3x⁵
   = 3x⁶
   which is evidently an even function.
4. .In symbols,
   f(x) = odd & g(x) = odd ⟹ f(x) ÷ g(x) = even
   For example, if f(x) = 3x⁵ and g(x) = x³, then
   f(x)/g(x) = 3x⁵/x³
   = 3x²
   which is also an even function.
5. The composition of two odd functions is always an odd function. In symbols,
   f(x) = odd & g(x) = odd ⟹ f∘g(x) = odd and g∘f(x) = odd
   For example, if f(x) = 2x and g(x) = x³, then
   f∘g(x) = f[g(x)]
   = 2x³
   which is obviously odd.Likewise,
   g∘f(x) = g[f(x)]
   = (2x)³
   = 8x³
   which is odd as well.
6. The composition of an even and an odd function is always even.In symbols,
   f(x) = odd & g(x) = even ⟹ f∘g(x) = even and g∘f(x) = even
   or
   f(x) = even & g(x) = odd ⟹ f∘g(x) = even and g∘f(x) = even
   For example, if f(x) = 2x² and g(x) = x³, then
   f∘g(x) = f[g(x)]
   = 2(x³)²
   = 2x⁶
   which is an even function.Likewise,
   g∘f(x) = g[f(x)]
   = (2x²)³
   = 8x⁶
   which is an even function as well.

You have reached the end of Math lesson 16.7.10 Properties 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

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1. Property Odd Feedback. Helps other - Leave a rating for this property odd (see below)
2. Functions Math tutorial: Even and Odd Functions. Read the Even and Odd Functions math tutorial and build your math knowledge of Functions
3. Functions Revision Notes: Even and Odd Functions. Print the notes so you can revise the key points covered in the math tutorial for Even and Odd Functions
4. Functions Practice Questions: Even and Odd Functions. Test and improve your knowledge of Even and Odd Functions with example questins and answers
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6. Continuing learning functions - read our next math tutorial: Relation and Function

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