Problem 69 - Algebra 2

Question

Open-Ended Write a function rule that approximates each value. a. The amount you save is a percent of what you earn. (You choose the percent.) b. The amount you earn depends on how many hours you work. (You choose the hourly wage.) c. Write and simplify a composite function that expresses your savings as a function of the number of hours you work. Interpret your results.

Step-by-Step Solution

Answer

The function rules are S(E) = 0.3E (savings as a function of earnings), E(H) = 20H (earnings as a function of hours worked), and S(H) = 6H (savings as a function of hours worked). This means that for every hour worked, you save $6.

Step 1: Write a function for savings

Let's say you save 30% of what you earn. This could be represented as a function: \S(E) = 0.3E \Where: S is the amount saved, E is the amount earned.

Step 2: Write a function for earnings

Let's say your hourly wage is $20. This could be represented as a function: \E(H) = 20H \Where: E is the amount earned, H is the number of hours worked.

Step 3: Write a composite function

The composite function that expresses your savings as a function of the number of hours worked, can be found by substituting the second function into the first. This gives: S(H) = S(E(H)) = 0.3 * E(H) = 0.3 * 20H = 6H

Step 4: Interpret the results

The function S(H) = 6H tells us that the amount saved is directly proportional to the number of hours worked. For each hour worked, you save $6.

Key Concepts

Function Rule

A "function rule" is a mathematical expression that defines how one quantity depends on another. In this specific exercise, we're interested in how savings relate to earnings and earnings relate to hours worked. By setting up a function rule, we can easily calculate these amounts.
For instance, if we save a percentage of what we earn, the function rule for savings could be defined as:

• \( S(E) = pE \)

Here, \( S(E) \) represents the savings, \( E \) the earnings, and \( p \) the percent of savings as a decimal. In our example, if \( p = 0.3 \) (meaning 30%), our function rule becomes \( S(E) = 0.3E \). This means for every dollar earned, 30 cents is saved.
Writing and understanding function rules are essential as they allow for quick and accurate calculations in various scenarios.

Percent Savings

"Percent savings" refers to the portion of your earnings that you choose to save, expressed as a percentage. Deciding on a suitable percentage for savings is crucial for financial planning and budgeting.
For example, if you choose to save 30% of what you earn, and your monthly income is $1000, then your savings will be \( 1000 \times 0.3 = 300 \) dollars.
This percentage can be adjusted based on your financial needs and goals:

• Higher percentages can lead to more substantial savings over time.
• Lower percentages may offer more immediate disposable income but may impact long-term savings.

By expressing savings as a percentage, it becomes easier to adjust savings based on changes in income.

Hourly Wage

An "hourly wage" is the amount of money earned for each hour of work. It's a common way to quantify earnings for jobs that pay by the hour rather than a fixed salary.
Understanding your hourly wage is important for calculating potential earnings.
If you earn $20 per hour, your earnings function can be represented as \( E(H) = 20H \) where \( H \) is the number of hours worked. This expresses earning a straightforward and predictable manner – work more hours, earn more money.
This simplifies the calculation of earnings and helps in budgeting and financial forecasting, as each additional hour of work directly increases total earnings by the hourly rate.

Interpretation of Results

"Interpretation of results" involves understanding what the outcome of the function means in real terms. For this exercise, the composite function \( S(H) = 6H \) gives us a clear picture of savings relative to hours worked.
The result \( S(H) = 6H \) suggests that for every hour worked, $6 is saved. This is derived from combining our function rules for savings and earnings, where saving 30% of 20 dollars per hour results in saving 6 dollars for each hour.
This kind of interpretation allows individuals to quickly assess how changes in work hours can directly impact their savings, providing valuable insights for personal financial planning.
Key takeaways:

• Direct proportionality between hours worked and amount saved.
• Potential to use the composite function for setting financial goals based on expected hours worked.