For his services, a private investigator requires a 500...

For his services, a private investigator requires a 500 dollar retainer fee plus 80 dollar per hour. Let $x$ represent the number of hours the investigator spends working on a case. a. Find a function $f$ that models the investigator's fee as a function of $x$. b. Find $f^{-1} .$ What does $f^{-1}$ represent? c. Find $f^{-1}(1220) .$ What does your answer represent?

For his services, a private investigator requires a 500 dollar retainer fee plus 80 dollar per hour. Let $x$ represent the number of hours the investigator spends working on a case.
a. Find a function $f$ that models the investigator's fee as a function of $x$.
b. Find $f^{-1} .$ What does $f^{-1}$ represent?
c. Find $f^{-1}(1220) .$ What does your answer represent?

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Key Concepts

Linear Function Modeling

This concept involves creating a function in the form f(x) = mx + b to represent a real-world situation. In the context of the problem, it models a service charge that includes a fixed initial fee and an additional cost that depends linearly on the number of service hours provided.

Inverse Functions

Inverse functions reverse the effect of the original function by swapping the roles of the inputs and outputs. In algebra, finding the inverse means solving for x in terms of y, such that if f(x) gives an output, then f?¹(y) returns the original input, allowing us to determine the required input for a desired output.

Interpreting Inverse Functions in Context

In practical applications, interpreting the inverse function means understanding what the computed input value represents physically or contextually. For example, in cost modeling, the inverse function helps determine the number of hours (input) needed for a given total fee (output), linking mathematical results back to real-world quantities.

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