## Calculus: Early Transcendentals 8th Edition

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 $m_0$).
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 $m_0$).