Question

How do you determine (algebraically) whether the function, $f(x) = \dfrac{1}{{2{x^4}}}$ is even, odd or neither?

Answer 1

A function is said to be even if $f( - x) = f(x)$ i.e. the negative sign is absorbed in this case
And it is considered to be odd if $f( - x) = - f(x)$ i.e. the negative sign may not be absorbed or remains the same. Examples of even functions are any function raised to an even number of powers like ${x^2}$ , ${x^4}$ , ${x^6}$ etc. Whereas, examples of odd functions include ${x^3}$ , ${x^5}$ etc.

Complete step-by-step solution:
The best way to find whether the given function is even or odd, find the value of the function with positive value of the variable and then with the negative value of the variable
$\Rightarrow f(x) = \dfrac{1}{{2{x^4}}}$
Now, substitute the negative value in the function,
$\Rightarrow f( - x) = \dfrac{1}{{2{{( - x)}^4}}}$
Since, there is even power on the negative sign, it will eventually become positive, hence, finally we will get
$\Rightarrow \dfrac{1}{{2{{( - x)}^4}}} = \dfrac{1}{{2{x^4}}}$
If we notice closely, the result of the positive and negative value, the result is same i.e. $\dfrac{1}{{2{x^4}}}$
Therefore, the given function is even. The graph continues to infinity.

Note: On observation, one can spot that these results are neither similar to those of the odd functions or those of the even functions. Even functions are always symmetric about the y-axis whereas odd functions are always symmetric about the origin, graphically.
Another important property of the exponents is used in this question i.e. the even power of a negative sign converts it into a positive coefficient. This can be understood by the basic rules of integers as $\left( - \right) \times \left( - \right) = \left( + \right)$
This shows that the pairing of the negative signs result in a positive coefficient. In case of even number of terms, pairs will be formed but in case of odd number of terms, one term will always be left and would retain the negative sign for the expression.
A function is neither even nor odd if the functions follow the following two conditions:
If the function is $f(x)$ then for it to be neither odd nor even, the following values are returned if the negative value of the variable is substituted
$f( - x) \ne f(x)$
And
$f( - x) \ne - f(x)$