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