To construct such a function $f$, we just need to choose a value for $f(x)$ from $\{1,2, \ldots, 10\}$ for each $x \in\{1,2, \ldots, 10\}$. But the condition that $f(i)+f(j)=11$ whenever $i+j=11$ means that $$\begin{aligned} f(10) & =11-f(1) \\ f(9) & =11-f(2) \\ \vdots & \\ f(6) & =11-f(5) \end{aligned}$$ This means that once we have chosen $f(1), f(2), f(3), f(4)$, and $f(5)$, the five remaining values of $f(6), f(7), f(8), f(9)$, and $f(10)$ are already determined. The answer is therefore just the number of ways to choose these first five values. Since there are 10 possibilities for each one, we get that the answer is $10^{5}=100000$.