Problem 86 - Algebra and Trigonometry

Question

Exchange Rates The relative value of currencies fluctuates every day. When this problem was written, one Canadian dollar was worth 1.0573 U.S. dollar. (a) Find a function \(f\) that gives the U.S. dollar value \(f(x)\) of \(X\) Canadian dollars. (b) Find \(f^{-1} .\) What does \(f^{-1}\) represent? (c) How much Canadian money would \(\$ 12,250\) in U.S. currency be worth?

Step-by-Step Solution

Answer

(a) \(f(x) = 1.0573x\); (b) \(f^{-1}(y) = \frac{y}{1.0573}\); (c) 11,585.90 CAD.

Step 1: Define the Function

We are given that one Canadian dollar is worth 1.0573 U.S. dollars. To find a function \(f\) that gives the U.S. dollar value \(f(x)\) of \(x\) Canadian dollars, we multiply \(x\) (the amount in Canadian dollars) by 1.0573. Hence, the function \(f(x)\) is defined as: \[ f(x) = 1.0573x \] This function tells us how many U.S. dollars \(x\) Canadian dollars are worth.

Step 2: Determine the Inverse Function

To find the inverse function \(f^{-1}\), which will give the Canadian dollar value for a given U.S. dollar amount, we need to solve the equation \(f(x) = y\) for \(x\). Thus, we have \(1.0573x = y\). We solve for \(x\) by dividing both sides by 1.0573:\[ x = \frac{y}{1.0573} \] The inverse function is \(f^{-1}(y) = \frac{y}{1.0573}\). This function represents the amount of Canadian dollars that is equivalent to \(y\) U.S. dollars.

Step 3: Convert U.S. Dollars to Canadian Dollars

To determine how much \\(12,250 in U.S. currency would be worth in Canadian money, we use the inverse function \(f^{-1}(y)\). Substitute \(y = 12,250\) into the inverse function:\[ f^{-1}(12,250) = \frac{12,250}{1.0573} \] Calculate the value:\[ f^{-1}(12,250) \approx 11,585.90 \] Thus, \\)12,250 in U.S. currency is worth approximately 11,585.90 Canadian dollars.

Key Concepts

Inverse Function in Currency Conversion

Understanding the concept of an inverse function is crucial when dealing with currency exchange rates. In the context of converting Canadian dollars to U.S. dollars, the function \( f(x) = 1.0573x \) helps determine how much Canadian dollars are worth in U.S. currency. However, if you need to convert U.S. dollars back to Canadian dollars, we need the inverse function, denoted as \( f^{-1}(y) \).
This inverse function essentially reverses the process; it takes an amount in U.S. dollars and converts it back to Canadian dollars. To find it, we originally set \( f(x) = y \) and solved for \( x \). This process leads us to \( x = \frac{y}{1.0573} \), indicating the function \( f^{-1}(y) = \frac{y}{1.0573} \).
This inverse approach allows us to work backward—turning a U.S. dollar value into its original Canadian dollar equivalent. So, remember, an inverse function is all about reversing an operation or process, making it a vital tool in currency conversion problems.

Currency Conversion Explained

Currency conversion is the process of determining the equivalent value of money in one currency relative to another. This operation becomes frequent when dealing with international transactions or travel. Here, we focus on converting Canadian dollars to U.S. dollars and vice versa.
When given an exchange rate, it indicates how much one unit of a currency (Canadian dollars, in our case) is worth in another currency (U.S. dollars). Taking the original exchange rate of 1 Canadian dollar = 1.0573 U.S. dollars, we can calculate values in U.S. dollars for any given amount in Canadian dollars using the function \( f(x) = 1.0573x \).
Conversely, when you need to convert U.S. dollars into Canadian dollars, it is essential to use the inverse function \( f^{-1}(y) = \frac{y}{1.0573} \). This illustrates that currency conversion is not only about multiplication but also involves division, depending on the direction of conversion.

Canadian Dollar to U.S. Dollar Exchange Rate

Exchange rates are dynamic and fluctuate based on market conditions. For instance, the conversion rate from Canadian dollars to U.S. dollars, at a given time might be 1.0573. This rate tells us that 1 Canadian dollar is valued at 1.0573 U.S. dollars.
Using this exchange rate, to convert Canadian dollars to U.S. dollars, we multiply the Canadian dollar amount by 1.0573. For example, \( \\(100 \text{ CAD} \) would convert to \( 100 \times 1.0573 = 105.73 \text{ USD} \).
To convert back, using the inverse or reciprocal rate, we divide the U.S. money amount by 1.0573. For example, \( \\)12,250 \text{ USD} \) becomes \( \frac{12,250}{1.0573} \), which is approximately \( 11,585.90 \text{ CAD} \). Understanding this bidirectional conversion process ensures you can navigate financial transactions involving Canadian and U.S. dollars efficiently.