Example 1: From a container filled with 20 liters of pure alcohol, 1 liter is poured out, then it is refilled with water. After pouring out 1 liter of the mixed solution and refilling with water again, this process continues. If after the $k$-th pour $(k \geq 1)$, a total of $x$ liters of pure alcohol have been poured out, and if after the $(k+1)$-th pour, a total of $f(x)$ liters of pure alcohol have been poured out, then the expression for the function $f(x)$ is $\qquad$ .
Solution: Since after the $k$-th pouring, a total of $x$ liters of pure alcohol have been poured, then when pouring alcohol for the $(k+1)$-th time, there are still $(20-x)$ liters of pure alcohol left in the container. The $(k+1)$-th pouring removes $\frac{1}{20}(20-x)$ liters of pure alcohol, so,
$$
f(x)=x+\frac{1}{20}(20-x)=1+\frac{19}{20} x(1 \leqslant x<20) .
$$

Key Points for Solving the Problem: The concept of a function has two basic components - the correspondence and the domain. Only by clearly defining both can one complete the process of establishing a function.