Question

How do you determine if \[f(x)=4{{x}^{3}}\] is an even or odd function?

Answer 1

Hint: This question is about even and odd functions.
Even function: The function f is said to be even if the equation \[f(x)=f(-x)\] holds for all the values of x such that x and –x are in the domain of f.
Odd function: The function f is said to be odd if the equation \[-f(x)=f(-x)\] holds for all the values of x such that x and –x are in the domain of f.
We are going to use them in this question.

Complete step by step answer:
Let us solve the question.
We have to check whether the function \[4{{x}^{3}}\] is even or odd.
As we know that when\[f(x)=f(-x)\], then the function f is an even function.
And when\[-f(x)=f(-x)\], then the function is an odd function.
That means the equation \[f(x)-f(-x)=0\] holds only for even function and the equation \[f(x)+f(-x)=0\] holds only for even function.
Now, let us check for\[f(x)=4{{x}^{3}}\].
As \[f(x)=4{{x}^{3}}\] ,
Then, \[f(-x)=4{{\left( -x \right)}^{3}}=4\times (-x)\times (-x)\times (-x)=-4{{x}^{3}}\]
Then, we can say that
\[f(x)=-f(-x)\]
\[\Rightarrow 4{{x}^{3}}=-\left( -4{{x}^{3}} \right)\]
\[\Rightarrow 4{{x}^{3}}=\left( 4{{x}^{3}} \right)\]
Hence, seeing from the above equation, it can be seen that \[f(x)\] is an odd function.

Note:
For solving this type of question, we should know about odd and even functions. There is an alternate method for knowing that if the function is odd or even. The function should only be polynomials, and then we can give a decision on them if they are odd or even. If in the polynomial, all the terms are of odd degree, then the function will be odd. If all the terms in the polynomial are of even degree, then the function will be even. And if the terms in the polynomial are a mixture of odd and even degree, then the function is neither even nor odd.