The above two solutions are incorrect.

That's this parabola with the dotted part discarded.
We want to find its inverse:



So f(x) is this graph:



We want to find the equation of its inverse, the blue graph below, 
which is the projection of f(x) across the "identity line", which is 
the green dotted line below through the origin that leans right at 
am angle of 45°, and that has equation y=x.



To find the equation for the graph of the blue curve,

  

Step 1. Change f(x) to y

   

Step 2. Interchange x and y

   

Step 3. Solve for y

Swap sides:
   

Use quadratic formula with a=1, b=2, c=-1-x








  

We must require that y > 0, and therefore



This tells us that we can only use the + sign:

Square both sides:

 

So equation of the blue curve is

     

Step 4.  Finally, change y to f-1(x)

   

Edwin