Galaxy Rotation Curves and Mass Discrepancy

[redtree]

I apologize for the simple question, but I am trying to understand the Mass Discrepancy-Acceleration Relation and its relationship to ##\mu(x)## (from https://arxiv.org/pdf/astro-ph/0403610.pdf).

The mass discrepancy, defined as the ratio of the gradients of the total to baryonic gravitational potential, can be described by a simple function of centripetal acceleration:

##D(x) = \frac{\Phi'}{\Phi'_{b}} ##

Where ##x = a/a0## and ##D(x)## is the inverse of the following equation:

##\mu(x) = \frac{x}{\sqrt{1+x^2}}##

It's not clear to me how ##D(x)## is the inverse of the equation ##\mu(x) = \frac{x}{\sqrt{1+x^2}}##.

For example, how would one substitute ##\mu(x)## for ##D(x)##?

 

[haruspex]

I apologize for the simple question, but I am trying to understand the Mass Discrepancy-Acceleration Relation and its relationship to ##\mu(x)## (from https://arxiv.org/pdf/astro-ph/0403610.pdf).

The mass discrepancy, defined as the ratio of the gradients of the total to baryonic gravitational potential, can be described by a simple function of centripetal acceleration:

##D(x) = \frac{\Phi'}{\Phi'_{b}} ##

Where ##x = a/a0## and ##D(x)## is the inverse of the following equation:

##\mu(x) = \frac{x}{\sqrt{1+x^2}}##

It's not clear to me how ##D(x)## is the inverse of the equation ##\mu(x) = \frac{x}{\sqrt{1+x^2}}##.

For example, how would one substitute ##\mu(x)## for ##D(x)##?

If it means merely that D() is the inverse function of ##\mu(x)## then that would be ##D(x)=\frac x{\sqrt{1-x^2}}##.