Given H = x^2 + 8 what is H^-1?

The notation ##H^-1## means the inverse function. The equation of the inverse function can be found by switching the ##x## and ##y## values inside the function.
##y = x^2 + 8##
##x = y^2 + 8##
##x – 8 = y^2##
##y = +-sqrt(x – 8)##
Hence ##H^-1(x) = +-sqrt(x- 8)##.
Hopefully this helps!