Functions, Domain and Range, and Reading Graphs in Context

What a function is

In applied mathematics we constantly describe how one quantity depends on another: how the cost of a phone plan depends on the data used, how the temperature of a coffee depends on time, how the height of a drone depends on how long it has been climbing. A function is a precise way to capture such a dependence. A function $f$ is a rule that assigns to each input value exactly one output value. We write $f(x)$ to mean "the output of $f$ when the input is $x$." The key requirement is exactly one output: if you put in the same input twice you must get the same answer both times.

The set of allowed inputs is the domain, and the set of resulting outputs is the range. In pure math the domain might be "all real numbers for which the formula makes sense," but in modeling the dom