Question

A function f is odd and f(x)=x^2 for all x>0. Sketch the graph of this function and then write the rule of this function as a single formula.

Answer 1

`f(x) =(x^3)/(|x|)`

Explanation:

Being an odd function means that
`f(-x) = -f(x)`.

Graphically, it means the function will be symmetric across the origin.

So, the graph will have to look like the shown graph.

This graph looks quite similar to the graph
`f(x) = x^3`, so we can use that as our parent function.

To get the graph to equal
`x^2` for x>0, we must divide
`x^3` by x.

However, this will simply result in the function
`f(x) = x^2`.

If we divide by the absolute value of x however, the graph will look as desired.