1. Given $f(x)=x^{2}, g(x)=-\frac{1}{2} x+5, g^{-1}(x)$ represents the inverse function of $g(x)$. Let
$$
F(x)=f\left(g^{-1}(x)\right)-g^{-1}(f(x)) .
$$

Then the minimum value of $F(x)$ is $\qquad$ .
$$
\text { II.1. } \frac{70}{3} \text {. }
$$
$$
\begin{array}{l}
\text { Given } f(x)=x^{2}, g(x)=-\frac{1}{2} x+5 \\
\begin{array}{l} 
\Rightarrow g^{-1}(x)=10-2 x . \\
F(x)=f\left(g^{-1}(x)\right)-g^{-1}(f(x)) \\
=(10-2 x)^{2}-\left(10-2 x^{2}\right) \\
=6\left(x-\frac{10}{3}\right)^{2}+\frac{70}{3} .
\end{array}
\end{array}
$$

Since $F(x)$ is a quadratic function defined on $R$, when $x=\frac{10}{3}$, $F(x)$ reaches its minimum value of $\frac{70}{3}$.