Problem 93 - A Graphical Approach to Precalculus with Limits

Question

Solve the following. Complete the following. (a) Write a function \(F\) that converts \(x\) miles to feet. (b) Write a function \(I\) that converts \(x\) feet to inches. (c) Write a formula for the composition \((I \circ F)(x)\) (d) Explain what \((I \circ F)(x)\) calculates.

Step-by-Step Solution

Answer

(a) \(F(x) = 5280 \, x\); (b) \(I(x) = 12 \, x\); (c) \((I \circ F)(x) = 63360 \, x\); (d) It calculates inches in \(x\) miles.

Step 1: Write the Function for Miles to Feet

To convert miles to feet, we need to use the conversion factor that 1 mile equals 5280 feet. Thus, the function \(F(x)\) that converts \(x\) miles to feet is given by:\[ F(x) = 5280 \, x \]

Step 2: Write the Function for Feet to Inches

To convert feet to inches, we use the conversion factor that 1 foot equals 12 inches. Thus, the function \(I(x)\) that converts \(x\) feet to inches is given by:\[ I(x) = 12 \, x \]

Step 3: Compose the Functions

The composition \((I \circ F)(x)\) means that we first apply the function \(F\) to \(x\) and then apply \(I\) to the result of \(F(x)\). This results in the formula:\[ (I \circ F)(x) = I(F(x)) = I(5280 \, x) = 12 \, (5280 \, x) \]Simplifying, we get:\[ (I \circ F)(x) = 12 \, \cdot \, 5280 \, x = 63360 \, x \]

Step 4: Explain the Composition

The function \((I \circ F)(x)\) calculates the total number of inches in \(x\) miles. It converts \(x\) miles first to feet, and then those feet to inches, effectively combining both conversions into a single step.

Key Concepts

Miles to Feet Conversion

Converting miles to feet is a fundamental process in unit conversions, especially within the imperial system. A mile is a longer unit commonly used, and breaking it down to smaller units like feet makes more precise measurements possible. In the metric of conversions, it's important to understand the set conversion factor, which in the case of miles to feet is 1 mile equals 5280 feet. Why 5280? Historically, a mile has been defined in various ways, but the international mile as we know it now is standardized at 5280 feet. When we wish to convert miles into feet, we multiply the number of miles by 5280. For instance, the conversion of 2 miles into feet would be calculated as:\[ F(x) = 5280 \, x \]\[ F(2) = 5280 \, \times \, 2 = 10560 \, \text{feet} \]This formula is essential in practical scenarios, often intersecting with various applications in science and engineering.

Feet to Inches Conversion

Now that miles have been converted to feet, the next step often involves converting feet to even smaller units: inches. This step is crucial for detailed measurement requirements. The conversion factor here is straightforward—1 foot equals 12 inches. This stems from the imperial system where base twelve is often used due to its divisibility by 2, 3, and 4. To perform the conversion, multiply the number of feet by 12. For example, converting 3 feet into inches yields:\[ I(x) = 12 \, x \]\[ I(3) = 12 \, \times \, 3 = 36 \, \text{inches} \]Such conversions are highly useful in everyday life, particularly in fields like construction and manufacturing where precise measurements are key to tasks.

Unit Conversion Functions

Unit conversion functions simplify the process of converting one unit of measurement to another. They are incredibly useful in calculations that require multiple conversions. In mathematical terms, these functions apply specific conversion factors to transform input values, allowing quick conversions without manual calculation steps. Working with functions like these often involves:- **Defining the conversion factor:** This is essential for ensuring accuracy in the conversion.- **Applying the factor to the unit:** Multiply the original measurement by the conversion factor.For example, in converting feet to inches, the function \( I(x) = 12 \, x \) serves this purpose effectively. These functions pave the way toward faster computations, just like automated tools do in various digital applications.

Precalculus Applications

In precalculus, understanding how to use and apply functions such as those for unit conversions prepares students for more advanced mathematical concepts. Composing functions is a common topic that demonstrates how one function can be applied to another to form a new function. For instance, the composition \[ (I \circ F)(x) = 63360 \, x \]demonstrates how the conversion from miles to inches is handled in a single calculation. Each step of the process—first converting miles to feet, then feet to inches—is represented within a unified function. This kind of composition is beneficial because:- It reduces complexity by combining operations.- It ensures consistency and minimizes errors in conversion.Precalculus applications of function composition like these assist students in developing logical reasoning and problem-solving skills that are essential in various scientific and engineering fields.