If $g$ interpolates the function $f$ in $x_0, \ldots x_{n-1}$ and if $h$ interpolates the function $f$ in $x_{1}, x_{2}, \ldots, x_n$, then $$ g(x) + \frac{x_0 - x}{x_n - x_0}\big[ g(x) - h(x)\big] $$ interpolates $f$ in $x_{0}, \ldots, x_{n}$.
I do not know how to show this though. The exercise notes that $g$ and $h$ need not be polynomials.
So if $g$ interpolates $f$ in $x_0, \ldots, x_{n-1}$, then $g(x_i) = y_i = f(x_i)$. Similar for $h$. The difference $g(x) - h(x)$ gives $g(x_n) - h(x_0)$ since they share every other point $x_i$ for $0 < i < n$. From here, I am not sure where to go.