startwith y = (x + 8) ^ 2 - 3
replace y with x and x with y to get x = (y + 8) ^ 2 - 3
add 3 to both sides of the equation to get x + 3 = (y + 8) ^ 2
take the square root of both sides of the equation to get plus or minus sqrt(x + 3) = y + 8
subtract 8 from both sides of the equation to get -8 plus or minus sqrt(x + 3) = y
your inverse equation is y = -8 plus or minus sqrt(x + 3)
if that inverse equation is correct, then (x,y) in the regular equation will be equal to (y,x) in the inverse equation.
this can be seen to be true in the following graph.
what is shown is that the original equation and the inverse equation are reflected about the line y = x as they should be.
the points of each are equidistant from the line y = x, which is shown by drawing the line y = -x + 20 that is vertical to the line y = x.
the points equidistant from the line y = x will be on the line y = -x + 20 as shown.
the point (-2.91, 22.91) and the point (22.91, -2.91) are equidistant from the line y = x.
the point (-14.09, 34.09) and the point (34.09, -14.09) are equidistant from the line y = x.
that confirms that the regular equation and the inverse equation are indeed inverse equations of each other.