Question

Health Care Suppose that \(P(x)\) represents the percentage of income spent on health care in year \(x\) and \(I(x)\) represents income in year \(x\). Find a function \(H\) that represents total health care expenditures in year \(x\).

Short answer

The function representing total health care expenditures in year \(x\) is \(H(x) = I(x) \times \frac{P(x)}{100}\).

Step-by-step solution

Understand the Given Functions

Identify the provided functions: - \(P(x)\) represents the percentage of income spent on health care in year \(x\). - \(I(x)\) represents income in year \(x\).

Express Percentage as a Decimal

Convert the percentage function \(P(x)\) to a decimal by dividing by 100: \[\frac{P(x)}{100}\]

Formulate the Total Expenditure Function

The total health care expenditure \(H(x)\) is the product of the income \(I(x)\) and the decimal form of the percentage \(P(x)\): \[H(x) = I(x) \times \frac{P(x)}{100}\]

Key concepts

income function

An income function, typically denoted as \(I(x)\), represents the income of an individual or organization in a specific year \(x\). Understanding the income function is crucial because it serves as a foundation for numerous financial and economic analyses, including our example related to health care expenditures.

In our exercise, the income function \(I(x)\) quantifies how much money is made each year. This function is vital when calculating expenditures and determining financial capabilities. It might be a simple constant, representing steady income, or a more complex function showing income growth over time. Both cases impact the final outcome of financial calculations.

• If \(I(x)\) is a constant, say \(I(x) = 50000\), the income remains the same every year.
• But if \(I(x)\) is a variable function, such as \(I(x) = 40000 + 2000x\), income increases as the years pass.

The form of \(I(x)\) directly impacts the calculation of expenditures, as we'll see in further sections.

percentage function

A percentage function, denoted as \(P(x)\), typically expresses a ratio out of 100, representing a proportion of something, such as income spent on a particular category like health care. In our context:

\(P(x)\) represents the percentage of income that is allocated to health care expenses in year \(x\).

• For example, if \(P(2020) = 10\), it means that 10% of the income for year 2020 is spent on health care.
• This percentage helps determine how much of the total income is directed towards specific needs and obligations.

However, to perform specific calculations, percentages must be converted to decimals. This conversion, done by dividing the percentage by 100, simplifies mathematical operations as it standardizes the value into more manageable forms. For instance:

\[ \frac{P(x)}{100} \]

Converts the percentage \(P(x)\) from its percentage form (like 10%) to a decimal (like 0.10).

multiplicative model

The multiplicative model is a method where we find the product of two functions or values to derive a solution. This model is particularly useful in economic and financial calculations.

In our health care expenditure scenario, we use a multiplicative model to combine the income function \(I(x)\) with the decimal form of the percentage function \(\frac{P(x)}{100}\). Here's how it's formulated:

\[ H(x) = I(x) \times \frac{P(x)}{100} \]
This equation breaks down as follows:

• \(I(x)\): Income in year \(x\) encapsulates the available financial resources.
• \(\frac{P(x)}{100}\): Proportions income percentage into its decimal form, facilitating the multiplication process.

Multiplying these values provides the total health care expenditure \(H(x)\) in year \(x\). For example, if in year 2020, the income \(I(2020) = 50000\) and the percentage \(P(2020) = 10\), the total health care expenditure is:

\[ H(2020) = 50000 \times \frac{10}{100} = 5000 \] Thus, by applying the multiplicative model, you can easily determine how much is spent on health care based on the total income and the percentage allocated towards it each year.