Problem 40 - Algebra Form and Function

Question

Find a formula for \(w\) by scaling the input of \(f\). Let \(f(t)\) give the number of liters of fuel oil burned in \(t\) days, and \(w(r)\) the liters burned in \(r\) weeks.

Step-by-Step Solution

Answer

Answer: The formula for \(w(r)\) is \(w(r) = f(7r)\), where \(f(t)\) represents the liters of fuel oil burned in \(t\) days.

Step 1: Convert weeks to days

In order to find a relationship between the input variables, we must first convert the time units of weeks to days. Since there are 7 days in a week, we can start by expressing \(r\) weeks as \(7r\) days.

Step 2: Relate the input variables

Now that we have converted the time units, we can relate the input variables \(t\) and \(r\). We know that \(r\) weeks is equivalent to \(7r\) days. Thus, we have the relationship: $$t = 7r$$

Step 3: Find the formula for w(r)

Finally, we can use the relationship between \(t\) and \(r\) to express \(w(r)\) in terms of \(f(t)\). Since \(t = 7r\), we can substitute this into the function \(f(t)\) to obtain: $$w(r) = f(7r)$$

Step 4: Final formula

The formula we derived for \(w(r)\) by scaling the input of \(f\) is: $$w(r) = f(7r)$$

Key Concepts

Input Transformation

Input transformation is a key step when working with functions, especially when you need to change the input variable to match different contexts or units of measurement.
For example, consider the task of finding a formula for the number of liters of fuel oil burned over a different timespan.
Given a function, such as \( f(t) \), which provides the number of liters burned in \( t \) days, you may want to express this in terms of \( r \) weeks instead. To do this, it's necessary to change the input from days to weeks.

• This involves recognizing the relationship: one week equals seven days.
• By transforming the input variable of weeks to days — specifically, converting \( r \) weeks into \( 7r \) days — you align the new input with the initial function \( f(t) \).
• Such transformation ensures that the function remains applicable to the new scenario without fundamentally altering its dependency on time.

This method of transforming the input allows us to maintain consistency across different measures, simplifying how the function is used in different contexts.

Conversion of Units

In mathematics and real-world applications, sometimes the problem requires dealing with different units of measurement.
A perfect illustration is converting weeks into days when you need to adjust a function to work with a new timeframe.

• Time conversion is a typical necessity in many areas, such as science, engineering, and economics, to name a few.
• For the function \( f(t) \) used in this exercise, originally modeled based on days, you’d use the fact that one week is composed of seven days to perform the conversion.

This makes it possible to

• avoid the confusion that differing units can create in calculations,
• ensure accurate representations in mathematical expressions,
• and preserve the integrity of function outputs when converted back and forth.

Handling units correctly means clear, precise function applications no matter the input units.

Function Composition

Function composition allows you to combine multiple functions into one. By using function composition, you can transform, scale, or adjust a function to suit a specific need or context.
In this exercise, the core task was to express the liters of fuel oil burned in terms of weeks by composing the function \( w(r) \) with \( f(t) \), which already describes the oil consumption in days.

• The essence of the method is plugging one function into another—in the equation \( w(r) = f(7r) \), you substitute \( 7r \) for \( t \) in \( f(t) \).
• This results in \( w(r) \) being expressed as if \( f \) was directly operating on weeks instead of days.

Function composition is useful for

• enhancing the utility of existing models,
• streamlining calculations, and
• simplifying complex problems by breaking them into interrelated component functions.

By composing functions, you have a powerful tool to adapt and extend functional relationships to new or varied inputs.