When you take inverse of a function, it just means you are reflecting it along the line $y = x$ .Reflection does not change the area under the curve. So area between two curves is same as the area between the inverses of two curves. So
Area between two curves = $\int_{0}^{1} g^{-1}(x)-f^{-1}(x) dx = \int_{0}^{1} g(x)-f(x) dx = (1/2)*(1/2 - 1/4) = 1/8$ .
So $C$ is the correct answer