Answer
The inverse function is given by
$$v(m)=c\sqrt{1-\frac{m_0^2}{m^2}}.$$
It says at what speed $v$ is its mass equal to $m$.
Work Step by Step
First, we have to note that the mass $m$ of the particle is a positive quantity and the speed $v$ is nonnegative. Squaring the given expression we get
$$m^2=\frac{m_0^2}{1-\frac{v^2}{c^2}}.$$
By further transformations
$$\frac{m_0^2}{m^2}=1-\frac{v^2}{c^2}\Rightarrow \frac{v^2}{c^2}=1-\frac{m_0^2}{m^2}\Rightarrow v^2=c^2\left(1-\frac{m_0^2}{m^2}\right).$$
Now remembering that $v\geq0$ and $c>0$ we have
$$v=c\sqrt{1-\frac{m_0^2}{m^2}}.$$
This function says, given the rest mass of the particle $m_0$, at what speed $v$ will its mass be equal to $m$.
