I am looking for suggestions about ways to introduce the preimage of a set under a function. My experience is that many students find it a confusing concept. The definition I use is as follows:
Suppose that $f:X\to Y$ is a function, and $A\subseteq Y$, then the preimage of $A$ under $f$ is $$f^{-1}[A]=\big\{x\in X:f(x)\in A\big\}\;.$$
Note that when teaching engineering students, I use the idea of a measuring device being a mapping from a physical state space $X$ to an observation space $Y$. You can then ask the question: Given a particular measurement/reading/observation, what physical states could have generated it? This approach is inspired by Steven M. LaValle's Sensing and Filtering: A Fresh Perspective Based on Preimages and Information Spaces.