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In this task students are asked to analyze a function and its inverse when the function is given as a table of values. In addition to finding values of the inverse function from the table, they also have to explain why the given function is invertible. This can be done in two ways, either by arguing that the given function is strictly increasing, and therefore every output will come from a unique input, or by examining the given values in the table and observing that each output is attained by a unique input.

This task illustrates that a function and its inverse reverse how we can look at a situation. The original function asks “How much rain has fallen after different amounts of time?” The inverse function asks “How long did it take for a certain amount of rain to fall?”.

This task could be used for instruction or as assessment. The context also lends itself to introduce the idea of inverse functions. In that case, one could ask if given an amount of rainfall it would be possible to determine how long it has been raining. If the task is used for assessment, the instructor may want to specify the domain more clearly.

Rainfall