Math Lesson 16.7.1 - Definition of Even Functions

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Welcome to our Math lesson on Definition of Even Functions, this is the first 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 Even Functions

From the previous tutorials of this math course, it is known that some functions have the same y-value for two different x-values. For example, in the quadratic function

we obtain two equal y-values for x = -1 and x = 4, as

and

Therefore, it is not uncommon in functions to find the same y-value for two different x-values. Some functions that have such a feature include quadratic functions, absolute value functions, fourth-order polynomial functions, etc.

There is a special case where a function possesses the above feature. It occurs when two x-values that are equidistant from the origin have the same corresponding y-value. Thesefunctions are called even functions.

By definition, even functions are those functions for which f(x) = f(-x) for every x.

For example, f(x) = x² - 1 is an even function. Let's confirm this claim by trying some pairs of opposite numbers.

Example 1

Find whether the following functions are even or not.

1. f(x) = x² - 5x + 3
2. g(x) = x³ - 4x
3. h(x) = 2x + 1
4. i(x) = -x² + 9

Solution 1

To disprove the evenness of a function it is sufficient to find two opposite x-values that give different y-results. Thus,

1. For x = -1 and x = 1 we have
   f(-1) = (-1)² - 5 · (-1) + 3
   = 1 + 5 + 3
   = 6 + 3
   = 9
   and
   f(1) = 12 - 5 · 1 + 3
   = 1 - 5 + 3
   = -4 + 3
   = -1
   Thus, since f(-1) ≠ f(1), the function f(x) is not even.
2. Let's try again x = -1 and x = 1. Thus,
   g(-1) = (-1)³ - 4 · (-1)
   = -1 + 4
   = 3
   and
   g(1) = 13 - 4 · 1
   = 1 + 4
   = 5
   Thus, since f(-1) ≠ f(1), the function g(x) is not even.
3. Now, let's solve for x = -2 and x = 2. We have
   h(-2) = 2 · (-2) + 1
   = -4 + 1
   = -3
   and
   h(2) = 2 · 2 + 1
   = 4 + 1
   = 5
   Thus, since h(-1) ≠ h(1), the function h(x) is not even.
4. Again, let's solve for x = -2 and x = 2. We have
   i(-2) = -(-2)² + 9
   = -4 + 9
   = 5
   and
   i(2) = -22 + 9
   = -4 + 9
   = 5
   Thus, since i(-2) = i(2), the function i(x) is even.

You have reached the end of Math lesson 16.7.1 Definition 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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