Why must the symbol +- be written in front of the radical to write the inverse of the function f(x)=x^2?

Here is why:

When we solve for the inverse function (see the particular step below;  when we have:  " y² = x " ;

We take the "square root" of EACH SIDE of the equation; to isolate "y" as a single variable on one side of the equation;

→  √(y²) = √x ;

→ |y| = |√x| ;

y =   ± √x ; 

Because when we take the square root — or any "even root", for that matter—we have two solutions: of a variable, we have TWO SOLUTIONS:  a positive value; and a negative values;  
 
→  since:  1)  a "negative value"; multiplied by a "negative value" ;                       equals a "positive value" ;  
                 →  and as such:  a "negative value" ;
multiplied by that same "negative value" ;
           {that is:  a "negative value", "squared (i.e "raised to the power of "2"} ; ,
                 →  results in a positive value ; 
→ and since:  
                2)  a "postive value";  multiplied by a positive value" ;                       equals a "positive value" ;  
                 →  and as such:  a "positive value" ;
multiplied by that same "positive value" ; 
           {that is:  a "positive value", "squared (i.e "raised to the power of "2"} ; , 
                 →  results in a "positive value";
→ and since:
                3)  any given integer, in it "positive value", squared (i.e. raised to the power of "2");  results in a "positive value" ; 
→and since:  
               4)  that same aforementioned integer; in its "negative value" form, squared (i.e. raised to the power of "2"); results in that same aforementioned "positive value". 

Note the following:


"Given the function:  "f(x) = x² " ;   Find the "inverse function" .

Let "y" = f(x) " ; 

and write as:  " f(x) = y = x² " ;  

→  " y = x² " ; 

→ Now, rewrite the equation;  replacing the "y" with "x" ;
 and replacing the "x" with "y" ; 

→  " x = y² " ;  

Now, rewrite the question; isolating "y" as a single variable;  
   with no coefficient (save for the "implied coefficient of "1" ) ; 

→  " x = y² " ;  

↔  " y² = x ; 

Now, take  the square root of EACH SIDE of the equation; 
    to isolate "y" on one side of the equation; 

→ √(y²)  = √x ; 

→ |y| = |√x| ; 

→ y =  ± √x .

Replace the "y" with " f ⁻¹(x)" ; to indicate that this the "inverse function" ; 

and write the "inverse function" :

→ " f ⁻¹(x) =  ± √x " ;