Problem 125 - College Algebra with Modeling and Visualization

Question

Converting Units \(\quad\) The tables represent a function \(F\) that converts yards to feet and a function \(Y\) that converts miles to yards. Evaluate each expression and interpret the results. $$ \begin{array}{rlllll} x \text { (yd) } & 1760 & 3520 & 5280 & 7040 & 8800 \\ F(x) \text { (ft) } & 5280 & 10,560 & 15,840 & 21,120 & 26,400 \end{array} $$ $$ \begin{array}{cccccc} x(m i) & 1 & 2 & 3 & 4 & 5 \\ Y(x)(y d) & 1760 & 3520 & 5280 & 7040 & 8800 \end{array} $$ (a) \((F \circ Y)(2)\) (b) \(F^{-1}(26,400)\) (c) \(\left(Y^{-1} \cdot F^{-1}\right)(21,120)\)

Step-by-Step Solution

Answer

(a) 10,560 ft; (b) 8800 yd; (c) 4 mi.

Step 1: Interpret Function Composition

To find \((F \circ Y)(2)\), we first need to compute \(Y(2)\). From the second table, \(Y(2) = 3520 \text{ yd}\). Next, use this result as the input for the function \(F\), which is provided in the first table. Thus, evaluate \(F(3520)\), which is 10,560 feet.

Step 2: Determine Inverse Function Evaluation

For \(F^{-1}(26,400)\), consult the table where \(F(x) = 26,400 \text{ ft}\) and identify its corresponding \(x\) or input value. From the first table, \(F(8800) = 26,400\), so \(F^{-1}(26,400) = 8800 \text{ yd}\).

Step 3: Evaluate Composition of Inverses

To find \((Y^{-1} \cdot F^{-1})(21,120)\), start by finding \(F^{-1}(21,120)\) from the first table. The corresponding \(x\) value for 21,120 feet is 7040 yards, thus \(F^{-1}(21,120) = 7040 \text{ yd}\). Next, use this 7040 yards in \(Y^{-1}\). From the second table, \(Y(4) = 7040 \text{ yd}\), so \(Y^{-1}(7040) = 4 \text{ mi}\).

Key Concepts

Inverse Functions

Inverse functions are a critical concept that simplifies the process of finding original inputs based on outputs of a function. Let's delve into how inverse functions work. Suppose we have a function \( f \) that maps an input to an output. The inverse of this function, denoted as \( f^{-1} \), takes the output of \( f \) and maps it back to the original input.

For example, given a function \( F \), if \( F(x) = y \), then \( F^{-1}(y) = x \). Essentially, applying an inverse function "undoes" the operation of the original function.

In the exercise, calculating \( F^{-1}(26,400) \) involved identifying the input value from the output of 26,400 feet. From the table provided, \( F(8800) = 26,400 \) means \( F^{-1}(26,400) \) returns 8800 yards. Inverse functions are not only useful in solving equations but also in real-world applications like reversing computations in engineering and physics.

Unit Conversion

Unit conversion is a practical application of mathematical functions that allows us to switch measurements between different units. This is incredibly useful in everyday situations like measuring distances, areas, or volumes without requiring complex systems.

To convert units, you typically use a conversion factor, which is often a multiplication or function, such as the function \( Y \) in the provided problem, that converts miles to yards. As seen in the tables, 1 mile converts to 1760 yards, making the function \( Y(x) = 1760 \times x \).

Similarly, converting yards to feet involves the function \( F \), where each yard equals 3 feet. Therefore, for instance, \( F(3520 \text{ yd}) \) converts to feet via multiplication, resulting in 10,560 feet. In practical use, these conversion functions help calculate distances for travel, determine fabric lengths, or assess land area sizes simply by changing the units.

Mathematical Tables

Mathematical tables are valuable tools for simplifying calculations. They list inputs and their corresponding outputs based on predefined mathematical rules or functions. They help visualize how inputs relate to outputs in functions, whether they involve linear transformations or more complex mappings.

In the original exercise, such tables were used to show conversions between miles, yards, and feet. For example, the table listing \( x(m \text{ i}) \) to \( Y(x)(y \text{ d}) \) shows that input values like 2 miles yield 3520 yards, reflecting the conversion rule applied via the \( Y \) function.

These tables facilitate quick reference without recalculating every time. They are helpful in contexts from physics—where tables might link temperature readings to pressure values—to chemistry, where reacting proportions are tabulated. In learning environments, they provide a clear, structured means of understanding function compositions and conversions by laying out data systematically. By using such tables, students enhance their comprehension of relationships inherent in functions.