Math Lesson 16.7.9 - Properties of Even Functions

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Welcome to our Math lesson on Properties of Even Functions, this is the ninth 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 Even Functions

If two even functions f(x) and g(x) are given, the following properties are true:

1. The sum of two even functions is always an even function.In symbols,
   f(x) = even & g(x) = even ⟹ f(x) + g(x) = even
   For example, if f(x) = 2x² and g(x) = x⁴, then
   f(x) + g(x) = 2x² + x⁴
   which is an even function as well, as it contains only even-degree terms.
2. The difference of two even functions is always an even function.
   In symbols,
   f(x) = even & g(x) = even ⟹ f(x) - g(x) = even
   For example, if f(x) = 3x⁴ and g(x) = x², then
   f(x)-g(x) = 3x⁴ - x²
   which is an even function as well, as it contains only even-degree terms.
3. The product of two even functions is always an even function.
   In symbols,
   f(x) = even & g(x) = even ⟹ f(x) ∙ g(x) = even
   For example, if f(x) = x² and g(x) = 3x⁴, then
   f(x) ∙ g(x) = x² ∙ 3x⁴
   = 3x⁶
   which is also an even function.
4. The quotient of division of two even functions is always an even function.
   In symbols,
   f(x) = even & g(x) = even ⟹ f(x) ÷ g(x) = even
   For example, if f(x) = 3x⁴ and g(x) = x², then
   f(x)/g(x) = 3x⁴/x²
   = 3x²
   which is also an even function.
5. The composition of two even functions is always an even function. In symbols,
   f(x) = even & g(x) = even ⟹ f∘g(x) = even and g∘f(x) = even
   For example, if f(x) = 4x² - 1 and g(x) = x⁴ + 2, then
   f∘g(x) = f[g(x)]
   = f(x⁴ + 2)
   = 4⋅(x⁴ + 2)²-1
   = 4⋅(x⁸ + 4x⁴ + 4)
   = 4x⁸ + 16x⁴ + 16
   It is evident that this function is even, as it contains only even-degree monomial terms.
   Likewise,
   g∘f(x) = g[f(x)]
   = g(4x² - 1)
   = 4⋅(4x² + 2)⁴ + 2
   We can use the help of the binomial coefficients calculator provided at the bottom of this page to obtain the following composite function
   g∘f(x) = 4⋅(256x⁸ + 512x⁶ + 256x⁴ + 128x² + 16) + 2
   = 1024x⁸ + 2048x⁶ + 1024x² + 512x² + 64 + 2
   = 1024x⁸ + 2048x⁶ + 1024x² + 512x² + 66
   It is clear that this function is even, as it contains only even-degree monomial terms.

You have reached the end of Math lesson 16.7.9 Properties of Even 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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Enjoy the "Properties of Even Functions" math lesson? People who liked the "Even and Odd Functions lesson found the following resources useful:

1. Property Even Feedback. Helps other - Leave a rating for this property even (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
5. Check your calculations for Functions questions with our excellent Functions calculators which contain full equations and calculations clearly displayed line by line. See the Functions Calculators by iCalculator™ below.
6. Continuing learning functions - read our next math tutorial: Relation and Function

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