Question

What is an odd function?

Answer 1

Definition of an odd function:

Any given function $f\left(x\right)$ will be an odd function when it follows $\mathbf{f}\mathbf{(}\mathbf{-}\mathbf{x}\mathbf{)}\mathbf{=}\mathbf{-}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ $\mathbf{\forall }\mathbf{}\mathbf{x}\mathbf{\in }\mathbf{Domain}$.

The graph of an odd function is symmetrical about the origin.

Example:

$f\left(x\right)=x^3$

$$\begin{gathered} f(-x)={(-x)}^3 \\ \Rightarrow =-x^3 \\ \Rightarrow =-f\left(x\right) \end{gathered}$$

So ,$f\left(x\right)=x^3$ is an odd function.

The graph of the function $f\left(x\right)=x^3$ is,

Hence, any function is an odd function when it satisfies the relation$f(-x)=-f\left(x\right)$.