Given function has the domain  |x| <= 2,  or  -2 <= x <= 2.

The plot of the function is shown in the Figure below.



             

                 Figure.  Plot y =  + x  (red line).
            Blue line is the tangent line.  Green line is a secant line.



As x grows starting from x = -2,  the term    also grows.

So, in vicinity of x= -2,  function    is the sum of two monotonic terms;
therefore, it is also monotonically risen function there.


Then it gets its maximum value between 2 and 3 in 'y' somewhere between 1 and 2 in 'x' 
and starts declining. It gets the value of 2 at x= 0  and  x= 2.


Our task is to find the maximum of the function.  As soon as we get it, we will solve the problem.


Seeking an elementary Algebra solution, let's introduce new variable 

     = t.    (1)


Then 

     = t-x,

    4 - x^2 = (t-x)^2             <<<---===  after squaring both sides in previous equation

    4 - x^2 = t^2 - 2tx + x^2,

    2x^2 - 2tx + (t^2-4) = 0.       (2)


Out goal is to find the maximum value of 't'.  


    +----------------------------------------------------------------------------------------------+
    |   Equation (1) and the plot in the Figure tell us that maximum value of 't' is achieved      |
    |   when two different possible roots 'x' of equation (1) merge into one single root.          |
    |                                                                                              |
    |   In terms of equation (2), it means that we want to consider equation (2) as an equation    |
    |   for 'x' with parameter 't' and want to find 't' from the condition of merging its x-roots. |
    +----------------------------------------------------------------------------------------------+


It means that we want to equate the discriminant of equation (2) to zero,
considering (2) as a quadratic equation for 'x'

    0 = d = b^2 - 4ac = (-2t)^2 - 4*2*(t^2-4) = 4t^2 - 8t^2 + 32 = -4t^2 + 32,

or

    4t^2 = 32  -->  t^2 = 32/4 = 8  -->  t = +/- .


The negative value for t is, obviously, irrelevant/extraneous, and we reject it.

The positive value t =   gives the maximum value of the given function.

So, the range of the given function is [ , ],  giving the ANSWER to the problem's question.