Question

Foreign Exchange Traders often buy foreign currency in the hope of making money when the currency's value changes. For example, on April \(10,2018,\) one U.S. dollar could purchase 0.8101 euro, and one euro could purchase 132.317 yen. Let \(f(x)\) represent the number of euros you can buy with \(x\) dollars, and let \(g(x)\) represent the number of yen you can buy with \(x\) euros. (a) Find a function that relates dollars to euros. (b) Find a function that relates euros to yen. (c) Use the results of parts (a) and (b) to find a function that relates dollars to yen. That is, find \((g \circ f)(x)\). (d) What is \((g \circ f)(1000) ?\)

Short answer

The function relating dollars to yen is \((g \circ f)(x) = 107.2173717x\). For 1000 dollars, you can get 107217.3717 yen.

Step-by-step solution

Define the function relating dollars to euros

Given that one U.S. dollar can purchase 0.8101 euro, we define the function \(f(x)\) as follows: \( f(x) = 0.8101x \). This function represents the number of euros you can buy with \(x\) dollars.

Define the function relating euros to yen

Given that one euro can purchase 132.317 yen, we define the function \( g(x) \) as follows: \( g(x) = 132.317x \). This function represents the number of yen you can buy with \(x\) euros.

Find the composite function relating dollars to yen

To relate dollars to yen, we need to find the composite function \((g \circ f)(x)\). This is done by substituting the function \(f(x)\) into the function \(g(x)\):\( (g \circ f)(x) = g(f(x)) \)Since \( f(x) = 0.8101x \), we substitute it into \( g(x) \):\( g(f(x)) = g(0.8101x) \)Because \( g(x) = 132.317x \), we get:\( g(0.8101x) = 132.317(0.8101x) = 107.2173717x \)Thus, the composite function is \( (g \circ f)(x) = 107.2173717x \).

Calculate \( (g \circ f)(1000) \)

To find how many yen you can buy with 1000 dollars, we use the composite function and substitute \( x = 1000 \):\( (g \circ f)(1000) = 107.2173717 \times 1000 = 107217.3717 \)Therefore, \( (g \circ f)(1000) = 107217.3717 \).

Key concepts

Composite Functions

Let’s start with composite functions, which are essential in various math and real-life applications. A composite function is essentially a combination of two functions where the output of one function becomes the input of another. In our context, we're working with currency conversions. Imagine you have a function for converting dollars to euros and another function for converting euros to yen. The composite function will give you a direct conversion from dollars to yen.

In mathematical terms, if you have two functions like we do in the exercise, namely, \(f(x) = 0.8101x\) and \(g(x) = 132.317x\), the composite function \( (g \circ f)(x) \) can be found by substituting one function into the other. The resulting function \( (g \circ f)(x) = 107.2173717x \) helps you convert dollars directly to yen.

Currency Conversion

Currency conversion is a significant skill, especially in global finance and travel. It involves changing one currency into another using a specific exchange rate. For example, on April 10, 2018, one U.S. dollar could purchase 0.8101 euros, and one euro could purchase 132.317 yen.

Using these exchange rates, we defined the functions in the exercise to convert dollars to euros and then euros to yen. These conversions are straightforward applications of multiplication: you multiply the amount of currency you have by the exchange rate.
For instance, if you have 1000 dollars and want to convert them to yen, you first convert them to euros, and then those euros are converted to yen.

Function Operations

Function operations allow you to perform arithmetic operations on functions, such as addition, subtraction, multiplication, and division. In the exercise, we focused on function composition, which is a type of multiplication: substituting one function into another. This is very useful in situations where you need to chain together multiple operations.

In our case, we took the function \(f(x) = 0.8101x\) that converts dollars to euros and substituted it into the function \(g(x) = 132.317x\) that converts euros to yen. The composite function \( (g \circ f)(x) \) then gives us a direct conversion from dollars to yen. This operation combines the two steps (dollars to euros, then euros to yen) into one, simplifying the conversion process.

Algebra Applications

Algebra isn't just for solving equations; it's highly practical in real-world scenarios. This exercise demonstrated a typical application in finance, where algebra helps us understand and predict currency conversions efficiently.

By defining our functions algebraically, \(f(x) = 0.8101x\) and \(g(x) = 132.317x\), we can easily manipulate them to form a composite function. Understanding how to form and solve these functions can help in various fields beyond finance, including engineering, computer science, and even biology.

In this exercise, we're not just converting currency; we're illustrating how algebra and function operations can model and solve real-world problems.