A logistics coordinator uses the function $f(x) = 8x + 5$ to calculate the total processing fee for a shipment based on its weight $x$. To build a tool that determines the weight based on a known fee, the coordinator must find the inverse function. Arrange the following steps in the correct order to find $f^{-1}(x)$ algebraically.

A billing specialist uses the function $f(x) = 8x + 5$ to calculate a customer's total service charge in dollars, where $x$ represents the number of labor hours. To determine the number of labor hours based on a known total charge, the specialist uses the inverse function. Which of the following is the correct inverse function $f^{-1}(x)$?

A shipping coordinator uses the function $f(x) = 8x + 5$ to calculate the total cost of a delivery based on the number of packages, $x$. To determine the number of packages when the total cost is known, the coordinator must use the inverse function. Match each algebraic expression with the correct step in the process of finding the inverse of $f(x) = 8x + 5$.

A payroll clerk uses the function $f(x) = 8x + 5$ to calculate an employee's total bonus in dollars, where $x$ represents the number of years of service. To find the inverse function for determining years of service from a known bonus, the clerk replaces $f(x)$ with $y$. True or False: The next algebraic step of interchanging the variables $x$ and $y$ results in the equation $x = 8y + 5$.

Intermediate Algebraic Steps in Inverse Derivation

A data analyst is working with a cost projection model represented by the function $f(x) = 8x + 5$. To create a reverse lookup formula that calculates the original input from a known output, the analyst begins by replacing $f(x)$ with $y$ to write the equation $y = 8x + 5$. According to the standard algebraic process for finding an inverse function, the immediate next step is to ____ the variables $x$ and $y$ before solving for the new $y$.