The intuitive understanding of $\sqrt{x}$ to model "inversely proportional / inverse square"? Can someone lay out what the concept I am trying to convey below in a more clear manner?
Square root functions are sometimes used to model non-linear relationships.  The value of a square root is "proportional" to the number whose root you're taking.  I think it's called "inversely proportional" or the inverse square law? 
For example, I think I've also it as part of a gravitational formula to show that gravity pull is related to the distance.  But, it's not linear.  The closer you are, the stronger it is?  Is this also called a proportional relationship?  
Or was it distance traveled is inversely proportional to the square root of the time? 
Another example related to curving an class test.  In the example below, the higher scores will not get increased by much.  A 100 stays the same.  And the lower the score, the more absolute points it gets increased.   Visually, the graph of the square root function levels off, so the highest rate of change is for lower numbers.
$$g(x)=10\sqrt{x}$$

What is the basic idea at play for the $\sqrt{}$ function?  I am not interested in calculations, but would like to be able to recognize situations where a "square root relationship" is applies and is appropriate.
 A: A linear (often called proportional) relationship between two quantities $x$ and $y$ means that $y = kx$. Notice this is a graph of a line. The $k$ is the proportionality between $x$ and $y$. It is symmetrical (in that $x = 1/k \cdot y$). It says that if $x$ increases by $1$, then $y$ increases by $k$. Alternatively, if you double $x$, then $y$ doubles as well, and vice-versa. This is very different from an inverse square relationship.
Two quantities $y$ and $x$ are often called inversely proportional if $y = k/x$. This is symmetric (it implies $x = k/y$). It means that if $x$ doubles, then $y$ halves, and vice-versa.
An inverse-square relationship is another relationship between $y$ and $x$. It is not symmetrical. So if $y=k/x^2$, then $x = \sqrt{k}/\sqrt{y}$, hence the connection the the square root function. Let us assume $y = k/(x^2)$. If $x$ is multiplied by $t$, then $y$ is affected by a factor of $1/t^2$. If $y$ is multiplied by $t$, then $y$ is affected by a factor of $1/\sqrt{t}$.
The inverse-square relationship comes up a lot in physics. It is simply something we observed and measured to a great deal of accuracy for many situations. To determine whether two quantities, again $y$ and $x$, are related in this way, or any other way, examine the effect that changing one has on the other. If changing $x$ by a certain factor has the affect of changing $y$ by the inverse of the square or the inverse of the square root of that factor, then they are related by some inverse-square law.
For example, let's examine whether distance traveled and time are related in this way when traveling at a constant speed. If you double the amount of distance you travel, logically it takes you twice as long -- this is a linear relationship, so $\text{distance} = k\cdot \text{time}$. That $k$ happens to be the speed at which you travel.
As a second example, let's see if the intensity, $I$, of a pulse of light is related to the distance, $d$, from its source (the point at which it was emitted). The pulse of light can be seen as a spherical shell traveling outward. The fundamental fact is that the total amount of energy carried stays the same. It follows that if you were to measure the energy delivered to a surface of area $A$ over a period of $t$ seconds, you should get the same result no matter $d$. So $I\cdot A\cdot t=E$. Set $A$ = $4\pi d^2$, the surface area of the shell, to get the total energy delivered over $t$ seconds. If one now doubles the distance, then $A$ increases by a factor of four, and thus the intensity must be multiplied by $1/4$ to retain the same $E$. Thus, we have that $I = k/d^2$, an inverse square law.
A: I want to clarify a few terms, as they can be ambiguous.

*

*Two quantities are called proportional to each other when they vary at the same time and are related by a constant multiplicative coefficient $$y=ax.$$ For example, the volume of water and its weight are proportional.


*Two quantites are inversely proportional when their product is constant, $$xy=a$$ also written $$y=\frac ax.$$ The speed of a car and the duration of the travel are inversely proportional.


*The quantities are quadratically related when one is proprortional to the square of the other. $$y=ax^2$$ For example, the height a dropped stone travels is a quadratic function of the time it takes.


*Now comes the hard part. The square root is the inverse function if the square: $$y=ax^2\leftrightarrow x=a\sqrt y$$
(constants left unspecified). Hence if a quantity is quadratically related to another, the second is "square-rootely" related to the first. But there is no term for this ! An example is the falling time as a function of the height.
Mind the trap: $\dfrac 1x$ is the (multiplicative) inverse of $x$, while $\sqrt x$ is the functional inverse of $x^2$.
