Question

If a function f is an odd function, then what type of symmetry does the graph of f have?

Short answer

The graph of function f when it is odd is symmetric about origin.

Step-by-step solution

Step 1. Define even function and odd function.

If a function fsatisfies$f\left(-x\right)=f\left(x\right)$ for every number x in it domain, then the function f is called an even function.

If a function fsatisfies$f\left(-x\right)=-f\left(x\right)$ for every number x in it domain, then the function f is called an odd function.

Step 2. Check a function if its odd.

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

role="math" localid="1645130185025" $$\begin{gathered} f\left(-x\right)={\left(-x\right)}^3 \\ ={\left(-1\right)}^3{x}^3 \\ =-\left(x^3\right) \\ =-f\left(x\right) \end{gathered}$$

Step 3. Interpretations.

The graph of the odd function starts at a point on one side of y axis and extend same length on the other side of the y axis.

Thus, the graph of an odd function is symmetric about the origin.