Question

You have to Determine analytically if the following function are even, odd, or neither.

Answer 1

Given the function

$f(x)=2x^3-x$

Using the graphical method,

The graph of the given function is shown below

From the graph of the function,

$f(-x)=-f(x)$

Hence, the given function is Odd

Alternatively

$f(x)=2x^3-x$

Replacing x with -x and solving

$\begin{gathered} f(x)=2x^3-x \\ \text{where x =-x} \\ f(-x)=2(-x)^3-(-x) \\ f(-x)=2(-x^3)+x=-2x^3+x \\ f(-x)=-2x^3+x=-1(2x^3-x) \\ f(-x)=-1(2x^3-x) \\ R\text{ecall that }f(x)=2x^3-x \\ \text{Thus} \\ f(-x)=-f(x) \end{gathered}$

It can be seen from the above deduction that

$f(-x)=-f(x)$

Hence, the function f(x)=2x³-x is odd