Problem 46 - Algebra 2

Question

According to the terms of Lavon’s insurance plan, he must pay the first \(\$ 300\) of his annual medical expenses. The insurance company pays 80\(\%\) of the rest of his medical expenses. Write a function for how muct the insurance company pays if \(x\) represents Lavon's annual medical expenses.

Step-by-Step Solution

Answer

The insurance payment function is \( f(x) = 0.8(x - 300) \) for \( x > 300 \), and \( f(x) = 0 \) for\( x \leq 300 \).

Step 1: Understand the Problem

Lavon has a deductible of \( \\(300 \). This means he has to cover the first \( \\)300 \) of his medical expenses himself. The insurance then covers 80% of any expenses above this amount.

Step 2: Identify the Variable

Let \( x \) represent Lavon's total annual medical expenses.

Step 3: Determine Expenses Above Deductible

If Lavon's total expenses \( x \) are more than \( \$300 \), then the insurance covers the amount \( x - 300 \).

Step 4: Calculate Insurance Responsibility

The insurance company pays 80% of the expenses above the \( \$300 \) deductible. Therefore, the amount paid by the insurance is\[ 0.8 \times (x - 300) \].

Step 5: Formulate the Function

The function representing the amount the insurance company pays, if \( x \) is Lavon's annual medical expenses, is \[ f(x) = \begin{cases} 0, & \text{if } x \leq 300 \ 0.8 \cdot (x - 300), & \text{if } x > 300 \end{cases} \]

Key Concepts

Deductible

A deductible is a set amount of money that an individual must pay out of pocket before their insurance starts to cover costs. In the context of insurance, it acts as a threshold that separates personal liability from what's covered by the insurer. For example, Lavon's insurance policy includes a deductible of \\(300. This means if Lavon incurs \\)500 in medical expenses in a year, he will pay the first \\(300 on his own. Only after this \\)300 is he eligible for insurance benefits to cover the remaining expenses. The deductible helps keep insurance premiums lower because it reduces the number of small claims an insurance company must process. It also encourages insured individuals to take preventive measures to avoid unnecessary expenses.

Annual Medical Expenses

Annual medical expenses are the total amount of money spent on medical needs over a year. This can include doctor visits, prescriptions, hospital stays, and other healthcare services. Understanding total annual expenses is crucial when dealing with insurance plans, as it helps in planning what out-of-pocket costs might be involved. Lavon's expenses need to account for everything he spends on healthcare throughout the year. If Lavon's expenses are below the deductible threshold of \\(300, he pays for all of it. When expenses exceed \\)300, his insurance begins to contribute towards covering those costs. Tracking annual medical expenses is vital for managing healthcare costs efficiently and ensuring one gets the most out of an insurance plan.

Piecewise Function

A piecewise function is a mathematical concept used to define a function with different expressions based on certain input intervals. This type of function is extremely useful for modeling scenarios like insurance payment where the payment changes based on expense levels. For Lavon's insurance, the payment function can be defined as:

• When Lavon's expenses, represented by \(x\), are \(x \leq 300\), the insurance payment is 0 because expenses haven't exceeded the deductible.
• When \(x > 300\), the insurance pays 80% of expenses over \$300, expressed as \[0.8 \times (x - 300)\]. The insurance company thus provides assistance only beyond the deductible threshold.

This method clearly distinguishes the expenses where the insurance does and does not make payments, simplifying the understanding of how coverage is applied. Piecewise functions allow for clear and concise descriptions of complex scenarios, demonstrating how different conditions affect outcomes.