Problem 81 - Algebra and Trigonometry

Question

Fee for Service For his services, a private investigator requires a \(\$ 500\) retention fee plus \(\$ 80\) 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?

Step-by-Step Solution

Answer

(a) \( f(x) = 80x + 500 \); (b) \( f^{-1}(x) = \frac{x - 500}{80} \); (c) 9 hours (work to charge $1220).

Step 1: Determine the function f(x)

The total fee the investigator charges is comprised of a flat retention fee plus an hourly rate. The retention fee is \(500 and the hourly fee is \)80 per hour. Therefore, the function that describes the fee as a function of hours is given by:\[ f(x) = 80x + 500 \]

Step 2: Find the inverse function f^{-1}(x)

To find the inverse function, we need to solve the equation \( y = 80x + 500 \) for \( x \). First, replace \( f(x) \) with \( y \):\[ y = 80x + 500 \]Subtract 500 from both sides:\[ y - 500 = 80x \]Divide both sides by 80 to solve for \( x \):\[ x = \frac{y - 500}{80} \]Thus, the inverse function is:\[ f^{-1}(x) = \frac{x - 500}{80} \]This inverse function \( f^{-1}(x) \) represents the number of hours required to reach a total fee of \( x \).

Step 3: Calculate f^{-1}(1220) and interpret

Use the inverse function to find \( f^{-1}(1220) \):\[ f^{-1}(1220) = \frac{1220 - 500}{80} = \frac{720}{80} = 9 \]This means that it took 9 hours of work for the total fee to be \(1220. Therefore, \( f^{-1}(1220) \) represents the number of hours the investigator needs to work to charge \)1220 in total.

Key Concepts

Inverse Function

An inverse function essentially reverses the roles of inputs and outputs in a given function. If you have a function that takes an input and provides an output, the inverse function does the opposite; it takes an output and provides the corresponding input.
The inverse function is useful when you know the result of a function and want to figure out what input produced that result. In our case with the private investigator's fee model, where the function is defined as \( f(x) = 80x + 500 \), the inverse function helps in determining the number of hours \( x \) needed to result in a certain fee.
To find this inverse, you need to solve the equation for \( x \) by rearranging terms. Once we solve \( y = 80x + 500 \) for \( x \), we get the inverse function:

• Subtract the flat fee from both sides to isolate the variable term: \( y - 500 = 80x \).
• Divide by the hourly rate: \( x = \frac{y - 500}{80} \).

The inverse function \( f^{-1}(x) = \frac{x - 500}{80} \) then tells us the number of hours needed for a particular fee.

Linear Function

Linear functions are mathematical expressions of the form \( f(x) = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. They describe relationships with constant rates of change, making them very predictable and easy to work with.
In the context of the investigator's fee model, the function is linear:

• The slope \( m = 80 \) represents the hourly rate. This means for every additional hour worked, the fee increases by \( $80 \).
• The y-intercept \( b = 500 \) represents the initial fee charged before any work is done, also known as the retention fee.

Linear functions like these are key in modeling straightforward relationships, allowing us to predict the outcome (fee) based on an input variable (hours worked). They form a straight line when graphed, hence the name "linear."

Function Modeling

Function modeling involves creating a mathematical representation of a real-world situation. It allows us to describe, analyze, and predict behaviors and outcomes.
For example, in the investigator’s fee problem, function modeling helps us formulate how fees relate to hours worked.

• The function \( f(x) = 80x + 500 \) models the private investigator's fees based on hours worked.
• By establishing this relation, we can easily calculate total charges for any number of hours.
• Similarly, using the inverse \( f^{-1}(x) = \frac{x - 500}{80} \), we can determine how long the work took for a given fee.

This use of function modeling is crucial in finance, engineering, science, and many other fields as it turns complex relations into clear mathematical statements, aiding in decision-making and problem-solving.

Hourly Rate Calculation

Hourly rate calculation is about determining how costs correlate with time, which is essential in various professional contexts, such as freelance work or consulting.
In our exercise, the private investigator's hourly rate is \( \(80 \). This means for every hour the investigator works, the fee charged increases by \( \)80 \).
This straightforward calculation is part of the linear function \( f(x) = 80x + 500 \), where multiplying the hourly rate by the number of hours and adding a flat fee provides total charges. These calculations help clients and providers understand cost structures and plan effectively based on expected work hours.

• Flat fee: \( \(500 \) is charged upfront, covering initial costs.
• Hourly rate: An additional \( \)80 \) is charged per hour worked.

Understanding hourly rate calculation is vital for budget management, offering guidance on what costs to expect depending on time spent on the job.