Math Lesson 16.7.5 - Proving the Oddness of a Function Analytically

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Welcome to our Math lesson on Proving the Oddness of a Function Analytically, this is the fifth 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.

Proving the Oddness of a Function Analytically

It is obvious that substituting certain x-values in a function's formula does not give us a definitive answer regarding the oddness of the given function. The best method for this is to check whether f(-x) = -f(x) for any x. In other words, we must check whether f(-x) - [-f(x)] = 0. If yes, the function is odd; otherwise, it is not odd. Let's explain this point through a few examples.

Example 5

Check, analytically, whether the following functions are odd or not.

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

Solution 5

1. For the function f(x) = x + 2x³ we have
   f(-x) = (-x) + 2 ∙ (-x)³
   = -x - 2x³
   and
   -f(x) = -(x + 2x³ )
   = -x - 2x³
   Therefore,
   f(-x) - [-f(x)] = (-x - 2x³ )-(-x - 2x³ )
   = -x - 2x³ + x + 2x³
   = 0
   Since the result of f(-x) - [f(x)] is zero, this means the function f(x) is odd.
2. For the function g(x) = x³ + 2 we have
   g(-x) = (-x)³ + 2
   = -x³ + 2
   and
   -g(x) = -(x³ + 2)
   = -x³ - 2
   Therefore,
   g(-x) - [-g(x)] = (-x³ + 2) - (-x³ - 2)
   = -x³ + 2 + x³ + 2
   = 4
   Since the result of g(-x) - [g(x)] is different from zero, this means the function g(x) is not odd.
3. For the function h(x) = -4x we have
   h(-x) = -4 ∙ (-x)
   = 4x
   and
   -h(x) = -(-4x)
   = 4x
   Therefore,
   h(-x) - [-h(x)] = 4x - 4x
   = 0
   Since the result of h(-x) - [h(x)] is zero, this means the function h(x) is odd.
4. For the function i(x) = 3x - 1 we have
   i(-x) = 3 ∙ (-x) - 1
   = -3x - 1
   and
   -i(x) = -(3x - 1)
   = -3x + 1
   Therefore,
   i(-x) - [-i(x)] = (-3x - 1) - (-3x + 1)
   = -3x - 1 + 3x - 1
   = -2
   Since the result of i(-x) - [i(x)] is different from zero, this means the function i(x) is not odd.

You have reached the end of Math lesson 16.7.5 Proving the Oddness of a Function Analytically. There are 10 lessons in this physics tutorial covering Even and Odd Functions, you can access all the lessons from this tutorial below.

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