A quality control technician uses a function $f$ to record the relationship between a machine's setting and the resulting output length in millimeters: $f = \{(0, 4), (1, 7), (2, 10), (3, 13)\}$. To determine which machine setting corresponds to a desired output length, the technician needs the inverse function $f^{-1}$. Which of the following sets represents the inverse function $f^{-1}$?

A telecommunications company uses a function $f = \{({}0, 4), ({}1, 7), ({}2, 10), ({}3, 13)\}$ to link the number of additional data lines a business has (input) to their monthly service fee in dollars (output). To find the number of lines based on a specific bill amount, the billing department uses the inverse function $f^{-1}$. Match each monthly bill amount (the input for $f^{-1}$) with the corresponding number of data lines (the output for $f^{-1}$).

An environmental agency uses a function $E = \{(0, 4), (1, 7), (2, 10), (3, 13)\}$ where the input is the number of hours a filtration system runs and the output is the number of pollutants removed in grams. To determine the hours needed for a specific amount of pollutant removal, the agency uses the inverse function $E^{-1}$.

True or False: The domain of the inverse function $E^{-1}$ is $\{4, 7, 10, 13\}$.

Inverse Function for Logistics Mapping

A manufacturing plant maps the number of hours a machine runs to the number of maintenance checks required using the function $M = \{(0, 4), (1, 7), (2, 10), (3, 13)\}$. To work backwards from the number of maintenance checks to find the hours run, the plant supervisor uses the inverse function $M^{-1}$. In the inverse function $M^{-1}$, if the input is 10, the corresponding output is ____.

Database Query Reversal for Budget Allocation

A logistics coordinator uses a scheduling system to map employee shifts (inputs) to the total number of packages processed (outputs) using the function $P = \{(0, 4), (1, 7), (2, 10), (3, 13)\}$. To analyze the reverse relationship, the coordinator needs to construct the inverse function $P^{-1}$, identify its domain, and identify its range.

Arrange the steps below in the correct order to successfully perform this inverse analysis, starting from identifying the original function and ending with defining the range of the inverse function.