Answer
The inverse function is
$$f^{-1}(m) = c\sqrt{1-\frac{m^2_0}{m^2}},$$
and it says at what speed of the body will its mass be equal to $m$ (given $m_0$).
Work Step by Step
To find the inverse function we have to solve the equation
$$m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$$
for $v$. By squaring we get
$$m^2=\frac{m_0^2}{1-\frac{v^2}{c^2}}\Rightarrow m^2\left(1-\frac{v^2}{c^2}\right)=m_0^2\Rightarrow 1-\frac{v^2}{c^2}=\frac{m_0^2}{m^2}\Rightarrow \frac{v^2}{c^2}=1-\frac{m_0^2}{m^2}\Rightarrow v^2=c^2\left(1-\frac{m_0^2}{m^2}\right)$$
now when we take the square root to get $v$ we only take the positive value because the speed is defined as the intensity of the velocity and thus is a positive value:
$$v=c\sqrt{1-\frac{m_0^2}{m^2}}.$$
This means that the inverse function is
$$f^{-1}(m) = c\sqrt{1-\frac{m^2_0}{m^2}},$$
and it says at what speed of the body will its mass be equal to $m$ (given $m_0$).
