Problem 75 - Precalculus : Building Concepts and Connections

Question

Applications In this set of exercises, you will use inverse functions to study real-world problems. A woman's dress size in the United States can be converted to a woman's dress size in France by using the function \(f(s)=s+30,\) where \(s\) takes on all even values from 2 to \(24,\) inclusive. (Source: www.onlineconversion \(. \operatorname{com})\) (a) What is the range of \(f ?\) (b) Find the inverse of \(f\) and interpret it.

Step-by-Step Solution

Answer

The range of function \(f(s)\) is the set of all even numbers from 32 to 54. The inverse of \(f(s)\) is given by \(f^{-1}(y) = y - 30\), and it converts a woman's dress size from French to US.

Step 1: Find the range of function \(f(s)\)

The range of a function is the set of all possible output values, in this case, French dress sizes. As \(s\) ranges from 2 to 24 in even increments, we just need to plug in these values into the function \(f(s) = s + 30\). For \(s = 2\), \(f(s) = 2 + 30 = 32\). For \(s = 24\), \(f(s) = 24 + 30 = 54\). Therefore, our range is all even numbers from 32 to 54, inclusive.

Step 2: Find the inverse of function \(f(s)\)

For finding an inverse, we switch the roles of \(s\) and \(f(s)\), and then solve for \(s\). Let's express \(f(s)\) as \(y = s + 30\). Setting \(s\) equal to \(y\) for the inverse gives us: \(s = y - 30\). This gives us the inverse function, \(f^{-1}(y) = y - 30\).

Step 3: Interpret the inverse function

Interpreting this inverse function, \(f^{-1}(y) = y - 30\), it is seen that this function converts a woman's dress size from French to US. If we input a French dress size \(y\), the output will be the corresponding US dress size.

Key Concepts

Function Range

Understanding the range of a function is key to many mathematical concepts and applications. Think of the range as the set of possible destinations you can reach with your function, which acts like a vehicle driving through the domain of inputs. In real-world terms, if we visualize a function as a vending machine, then the range would be the selection of snacks we get by pressing the buttons, which represent the inputs or domain.

Let's consider our example with the dress sizes. We're given a function that describes how U.S. dress sizes are converted to French dress sizes: by adding 30 to the U.S. size. The U.S. sizes go from 2 to 24, so, applying the function, the resulting French sizes—the output—range from 32 to 54. This practical visualization helps illustrate why we say the range of the function here is all even numbers from 32 to 54. It's as if the vending machine only offers snacks numbered from 32 to 54 when you press the even-numbered buttons.

Real-World Applications of Mathematics

When delving into the real-world applications of mathematics, we encounter countless ways that math serves as the foundation for problem-solving and decision-making. A fundamental aspect of these applications involves understanding and manipulating functions. In the context of our dress size conversion, mathematics offers a tool to standardize sizes across different regions, facilitating international shopping and manufacturing.

Real-world math shines when it simplifies everyday tasks, such as converting currency, cooking, or navigating traffic through the use of mathematical functions. These examples use functions to model relationships between different quantities, allowing us to make predictions and decisions. Another illustration would be architects using functions to calculate materials needed for building a house or economists predicting market trends. Mathematics thus becomes an indispensable part of our daily life.

Solving Inverse Functions

Solving inverse functions is like learning a secret handshake that lets you go backward in a mathematical process—undoing what you just did. It's the function's way of saying, 'I can take you back to where we started.' We apply this when we need to revert from the output back to the input.

In the context of the dress size problem, the inverse function helps us convert French dress sizes back to U.S. sizes. The process is straightforward: let's switch the output and input, solve for the new input, and there you have it, the inverse function. Specifically, the original function took a U.S. size, added 30, and gave us the French size. The inverse function takes the French size and subtracts 30 to return to the U.S. size. It reminds us that functions can be reversible if there's a one-to-one correspondence between the inputs and outputs, which is a powerful concept in mathematics.