Math Lesson 16.7.8 - What If a Function is Neither Even Nor Odd?

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Welcome to our Math lesson on What If a Function is Neither Even Nor Odd?, this is the eighth 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.

What If a Function is Neither Even Nor Odd?

Most functions are neither even nor odd. If the question requires studying a function, one of the elements to study involves checking the evenness and the oddness of that function. Let's consider an example in this regard.

Example 8

Classify the following functions into even, odd or neither even nor odd by checking their evenness/oddness analytically.

1. f(x) = log (2x)
2. g(x) = 2(x³ - x)²
3. h(x) = (x5 - x³) / x
4. i(x) = x²(x² - 1)

Solution 8

1. The function f(x) = log (2x) is defined only in the positive part of the X-axis. Therefore, there is no need to discuss f(-x). This means the function cannot be even, as we can find f(x) for any x > 0 but not the values of f(-x).
   The same reasoning is also used to exclude the possibility for the function f(x) to be odd given the condition f(-x) = -f(x).
   Therefore, the function f(x) is neither even, nor odd.
2. We have
   g(x) = 2(x³ - x)²
   = 2(x⁶ - 2x⁴ - x²)
   = 2x⁶ - 4x⁴ - 2x²
   In addition,
   g(-x) = 2 ∙ (-x)⁶ - 4 ∙ (-x)⁴ - 2 ∙ (-x)²
   = 2x⁶ - 4x⁴ - 2x²
   This function is identical to g(x), which means that g(x) is even. Indeed,
   g(x) - g(-x) = (2x⁶ - 4x⁴ - 2x²) - (2x⁶ - 4x⁴ - 2x²)
   = 2x⁶ - 4x⁴ - 2x² - 2x⁶ + 4x⁴ + 2x²
   = 0
   Now, let's check the oddness of the function g(x). We have
   -g(x) = -(2x⁶ - 4x⁴ - 2x²)
   = -2x⁶ + 4x⁴ + 2x²
   Thus,
   g(-x) - [-g(x)] = (2x⁶ - 4x⁴ - 2x²) - (-2x⁶ + 4x⁴ + 2x²)
   = 2x⁶ - 4x⁴ - 2x² + 2x⁶ - 4x⁴ - 2x²
   = 4x⁶ - 8x⁴ - 4x²
   Since the above operation gives a result that is different from zero, the function g(x) is not odd.
3. First, it is worth recalling that the function h(x) is defined for x ≠ 0, given that no number can be divided by zero in R. Thus, for x ≠ 0 we have
   h(x) = x⁵ - x³/x = x⁵/x - x^3/x
   = x⁴ - x²
   Now, let's check the evenness of this function. Thus,
   h(-x) = (-x)⁴ - (-x)²
   = x⁴ - x²
   It is identical to h(x), so the function h(x) is even. Indeed,
   h(x) - h(-x) = (x⁴ - x²) - (x⁴ - x²)
   = x⁴ - x² - x⁴ + x²
   = 0
   Hence, since h(x) - h(-x) = 0, the function h(x) is even.
   As for the oddness of h(x), we have
   -h(x) = -(x⁴ - x²)
   = -x⁴ + x²
   Hence,
   h(-x) - [-h(x)] = (x⁴ - x²) - (-x⁴ + x²)
   = x⁴ - x² + x⁴ - x²
   = 2x⁴ - 2x²
   Since the result of h(-x) - [-h(x)] is different from zero, the function h(x) is not odd.
4. We can express the function i(x) = x²(x² - 1) as
   i(x) = x⁴ - x²
   To prove the evenness of i(x) we must find i(-x) and subtract the two functions. If the difference is zero, the function is even. Thus, we have
   i(-x) = (-x)⁴ - (-x)²
   = x⁴ - x²
   Hence,
   i(x) - i(-x) = (x⁴ - x²) - (x⁴ - x² )
   = x⁴ - x² - x⁴ + x²
   = 0
   Therefore, the function i(x) is even.
   As for the oddness, we know that the function i(x) is odd if i(-x) = -i(x), i.e. if i(-x) - [-i(x)] = 0. In the specific case, we have
   i(-x) = x⁴ - x²
   and
   -i(x) = -(x⁴ - x²)
   = -x⁴ + x²
   Thus,
   i(-x) - [-i(x)] = (x⁴ - x²) - (-x⁴ + x² )
   = x⁴ - x² + x⁴ - x²
   = 2x⁴ - 2x²
   Since the above operation gives not zero, the function i(x) is not odd.

Remark! There is another category of functions called trigonometric functions, for which the evenness/oddness is very important. We explain this property (evenness/oddness) when working with trigonometric functions in chapter 23.

You have reached the end of Math lesson 16.7.8 What If a Function is Neither Even Nor Odd?. 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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