How do I find the function that is perfectly between y = x^2 and y = x? At first, I thought it was as simple as taking both functions adding them together then dividing by two, but this is not the case for what I am looking for. 
Here is a plot of the following:
y = x
y = x^2
y = 1/2*(x^2+x)
points exactly in between y = x and y = x^2

As you can see the red line representing y = 1/2*(x^2+x) does not land on the green points which are exactly in between the two functions y = x and y = x^2. What I am trying to learn how to do is figure out how to find the function which represents the exact middle between the two equations y = x and y = x^2.
I have already tried using an excel sheet to fit a line to all the green points I have calculated and still can't come up with a good line that fits.
I have looked into calculating the midpoint between two given points and that only helped calculate the points between the two equations, and didn't help towards obtaining a function that perfectly represents the line between y = x and y = x^2.
Thanks for any help or suggestions towards the right domain of math reserved for solving cases like this one.
cheers!!
 A: The green dots in your plot have the coordinates
a = {{0, 0}, {1, 1}, {3, 4}, {6, 9}, {10, 16}, {15, 25}, {21, 36}, {28, 49}, {36, 64}, {45, 81}}

which can be calculated with the formula
f[x_] = (1 + 4 x - Sqrt[1 + 8 x])/2

Test:
f[10]
(*    16    *)
f[45]
(*    81    *)

How to find this formula: in Mathematica,
FindSequenceFunction[a, x]
(*    (-1 + 1/2 (1 + Sqrt[1 + 8 x]))^2    *)

A: [A simple way to derive the function in the accepted answer.] It looks like you’re defining “exactly in between” as being halfway between the two graphs along a a horizontal line. That is, the $x$-coordinate of the in-between point is the average of the values of $x$ that produce the same value of $y$ for the two functions. Inverting the two functions, we then have for this midpoint $$x = \frac12(y+\sqrt y).$$ Solving for $x$ and taking the positive branch gives $$y = \frac12(1+4x+\sqrt{1+8x}).$$ 
A: The point between two points is simply $$\frac{1}{2} \left(p_1+p_2\right)=p_m$$
For example:
$$\frac{1}{2} (\{1,2\}+\{5,6\})=\{3,4\}$$
So an easy solution is to write a mean of the functions your have and we should get something in the middle. You say a line...but I assume you mean a curve?
f[x_] := Mean[{x^2, x}]
Plot[{x, x^2, f[x]}, {x, -10, 10}, ImageSize -> Large, PlotLegends -> "Expressions"]


There are no perfectly straight lines you can build between x and x^2, x^2 grows quadratically and x linearly, to stay perfectly on the mean of the two, you will end up building a curve and not a line.  
