Question

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

Answer 1

According to given in the question we have to determine that the given function which is $f(x) = \dfrac{1}{{2{x^4}}}$ even, odd or neither. So, as it is mentioned that we have to check algebraically so, we have to substitute any variable to check for the given function by observing its solution that the function obtained is even or odd.
Now, we have to substitute the value of –x in the place of x in the function which is as given $f(x) = \dfrac{1}{{2{x^4}}}$ and if it becomes positive or gives us the solution in the form of positive variable or function then we can say that the given function is even.
Now, we have to substitute the value of -x in the place of x in the function which is as given $f(x) = \dfrac{1}{{2{x^4}}}$ and if it becomes negative or gives us the solution in the form of negative variable or function then we can say that the given function is odd.
Now, we have to substitute the value of -x in the place of x in the function which is as given $f(x) = \dfrac{1}{{2{x^4}}}$ and if it becomes neither positive nor negative or gives us the solution in the form of neither positive nor negative variable or function then we can say that the given function is neither odd nor even.

Complete step-by-step answer:
Step 1: Now, we have to substitute the value of –x in the place of x in the function which is as given $f(x) = \dfrac{1}{{2{x^4}}}$ and if it becomes positive or gives us the solution in the form of positive variable or function then we can say that the given function is even. Hence,
$\Rightarrow f( - x) = \dfrac{2}{{{{( - x)}^4}}} \\\ \Rightarrow f( - x) = \dfrac{2}{{{x^4}}} \\\$
Hence, $f( - x) = f(x)$ which shows that the given function is an even function.
Step 2: Now, we have to substitute the value of -x in the place of x in the function which is as given $f(x) = \dfrac{1}{{2{x^4}}}$ and if it becomes negative or gives us the solution in the form of negative variable or function then we can say that the given function is odd. Hence,
$\Rightarrow f( - x) = \dfrac{2}{{{{( - x)}^4}}} \\\ \Rightarrow f( - x) = \dfrac{2}{{{x^4}}} \\\$
Hence, $f( - x) \ne - f(x)$ which shows that the given function is an even function.
Step 3: Now, we have to substitute the value of -x in the place of x in the function which is as given $f(x) = \dfrac{1}{{2{x^4}}}$ and if it becomes neither positive nor negative or gives us the solution in the form of neither positive nor negative variable or function then we can say that the given function is neither odd nor even.
$\Rightarrow f( - x) = \dfrac{2}{{{{( - x)}^4}}} \\\ \Rightarrow f( - x) = \dfrac{2}{{{x^4}}} \\\$
Hence, $f( - x) = f(x)$ which shows that the given function is an even function.

Final solution: Hence, we have determined that the given function $f(x) = \dfrac{1}{{2{x^4}}}$ is an even function.

Note:
To determine algebraically that the given function is even or odd it is necessary that we have to substitute the negative value of the variable in the place of the variable which is given in the function and if the function obtained is positive then we can say that the function is an even function and if the function obtained is negative then we can say that the given function is odd otherwise neither even nor odd.
If the function becomes positive or gives us the solution in the form of a positive variable or function then we can say that the given function is even.