Problem 71 - Algebra and Trigonometry

Question

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) \( f(x) = 500 + 80x \); (b) \( f^{-1}(y) = \frac{y - 500}{80} \); (c) 9 hours.

Step 1: Determine the Function

The investigator charges a \(500 retention fee plus \)80 per hour. This can be represented by the function \( f(x) = 500 + 80x \), where \( x \) is the number of hours worked.

Step 2: Verify the Function

Check that the function \( f(x) = 500 + 80x \) correctly calculates the fee. For instance, if \( x = 0 \), \( f(0) = 500 \), confirming a base fee of \(500. Similarly, for \( x = 1 \), \( f(1) = 580 \), which is correct as the fee is \\)500 + 1 \times 80\$80.

Step 3: Find the Inverse Function

To find the inverse function \( f^{-1} \), solve the equation \( y = 500 + 80x \) for \( x \).1. Subtract 500 from both sides: \( y - 500 = 80x \).2. Divide both sides by 80: \( x = \frac{y - 500}{80} \).Thus, the inverse function is \( f^{-1}(y) = \frac{y - 500}{80} \).

Step 4: Interpret the Inverse Function

The inverse function \( f^{-1}(y) \) represents the number of hours \( x \) required to achieve a fee \( y \). It essentially answers the question: "For a given fee, how many hours were worked?"

Step 5: Substitute into the Inverse Function

To find \( f^{-1}(1220) \), substitute 1220 for \( y \) in \( f^{-1}(y) = \frac{y - 500}{80} \).\( f^{-1}(1220) = \frac{1220 - 500}{80} = \frac{720}{80} = 9 \).

Step 6: Interpret the Result

The value \( f^{-1}(1220) = 9 \) means that the investigator worked 9 hours to accumulate a fee of $1220.

Key Concepts

Function Modeling

Modeling functions is like creating a map for real-world situations using mathematical equations. Consider this situation: a private investigator charges a base fee and an hourly rate. You can think of this as a scenario that changes based on the hours worked. To express this change through a function, we can establish a relationship between hours worked and total fees.

In our case, the function that models the investigator's fee is

• The base fee is \(\\(500\). This is charged regardless of hours worked.
• The hourly fee is \(\\)80\). This multiplies by the hours worked \(x\).
• Thus, the function is \(f(x) = 500 + 80x\). Here, \(f(x)\) represents the total fee, where 500 accounts for the base and \(80x\) adds the cost for each hour.

Function modeling helps visualize and calculate scenarios like these. It's particularly useful in predicting outcomes, such as total costs based on variable elements like time.

Cost Function

In mathematics, a cost function represents the total cost of providing a service based on a variable. In our exercise, the private investigator's fee can be seen as a cost function. This is a linear function used to express the relationship between hours worked and the total fee.

The cost function is given by:

• \(f(x) = 500 + 80x\) where \(x\) is the number of hours.

Let's analyze it further:

• The '500' in the function is a constant term. It represents the fixed cost for engaging the investigator.
• The '80x' term represents the variable cost. It depends on how many hours the investigator works.

Understanding cost functions enables predicting expenses. Also, it helps in making strategic decisions, like budgeting hours to fit a specific budget.

Algebraic Equations

Algebraic equations are mathematical statements that reflect the equality between two expressions. Solving them is essential in finding unknown values. In the context of our problem, the use of algebraic equations allows us to find the inverse function and answer specific questions about hours and fees.

Here's how we determine the inverse function:

• We start with the function: \(y = 500 + 80x\).
• To find the inverse, solve for \(x\). First, subtract 500 from both sides, leading to \(y - 500 = 80x\).
• Then, divide by 80: \(x = \frac{y - 500}{80}\).

The inverse function \(f^{-1}(y) = \frac{y - 500}{80}\) allows us to determine the number of hours worked based on a total fee \(y\). When we substitute \(y = 1220\), we find that \(f^{-1}(1220) = 9\), indicating that 9 hours were worked to reach a fee of \$1220. Understanding algebraic equations like this is crucial for solving problems and gaining insights into relationships within a function.