InverseFunction must be used with circumspection, because it is not well defined, and it has no Reals option.
It seems that Reduce is wrong in your particular case too. Remove for simplicity the ConditionalExpression:
g[x_] = x + (-70 + 21 x^2 +
Sqrt[7] Sqrt[
700 + 3 x (-800 + 3 x (300 + x (-80 + 7 x)))])/(-140 +
42 x) - (x (-70 + 21 x^2 +
Sqrt[7] Sqrt[
700 + 3 x (-800 + 3 x (300 + x (-80 + 7 x)))]))/(-140 +
42 x);
Then the output of the following is wrong in Mma 11.0.1:
Reduce[y == g[x], x, Reals]
You can check it with a plot:
h[y_] = Piecewise[{{Root[-700 y +
700 y^2 + (1300 - 490 y - 420 y^2) #1 + (-2190 + 840 y +
63 y^2) #1^2 + (1920 - 399 y) #1^3 + (-804 + 63 y) #1^4 +
117 #1^5 &, 1],
y < 1}, {Root[-700 y +
700 y^2 + (1300 - 490 y - 420 y^2) #1 + (-2190 + 840 y +
63 y^2) #1^2 + (1920 - 399 y) #1^3 + (-804 + 63 y) #1^4 +
117 #1^5 &, 2], y >= 1}}];
ParametricPlot[{h[y], y}, {y, -1, 2}]
