Problem 117 - Precalculus : Building Concepts and Connections

Question

The exchange rate from U.S. dollars to euros on a particular day is given by the function \(f(x)=0.82 x,\) where \(x\) is in U.S. dollars. If GlobalEx Corporation has revenue given by the function \(R(t)=40+2 t,\) where \(t\) is the number of years since 2003 and \(R(t)\) is in millions of dollars, find \((f \circ R)(t)\) and explain what it represents. (Source: www.xe.com)

Step-by-Step Solution

Answer

Hence, GlobalEx Corporation's annual revenue in millions of Euros as a function of time (years since 2003) is \( (f \circ R)(t) = 32.8 + 1.64t \).

Step 1: Understanding the Functions

Firstly, understand the given functions. \(f(x)=0.82 x\) is the conversion rate function for converting USD to Euro, and \(R(t)=40+2 t\) is the annual revenue function of GlobalEx Corporation in millions of USD.

Step 2: Computing the Composite Function

To find the composite function, \( (f \circ R)(t) \), replace \( x \) in \( f(x) \) with \( R(t) \) from the revenue function. This yields, \( (f \circ R)(t) \) = \( f(R(t)) \) = \( f(40+2t) \).

Step 3: Applying the Conversion Rate

Then apply the conversion rate to the revenue which results in \( f(40+2t) = 0.82*(40+2t) = 0.82*40+0.82*2t = 32.8 + 1.64t \) Euro.

Step 4: Interpretation

\( (f \circ R)(t) = 32.8 + 1.64t \) represents the annual revenue of GlobalEx Corporation in millions of Euros as a function of time (years since 2003).

Key Concepts

Function Composition

Understanding function composition is crucial in precalculus and beyond, as it involves creating a new function by combining two or more functions. Think of it as if one function 'feeds' its output into the next function as the input. The general notation for composing two functions, say function f and function g, is written as \( (f \circ g)(x) \), meaning you first apply g to x, then take that result and apply f.

From a real-world perspective, let's say you have the price of an item in one currency, and you need to find its price in another currency. You would first use a function to convert the currency, and then perhaps another function to apply a discount. This sequential processing is what function composition is all about. In the exercise given, we have composed the revenue function \( R(t) \) with the exchange rate function \( f(x) \) to find the revenue in Euros instead of U.S. dollars. This kind of practical application demonstrates the power of function composition in linking different mathematical concepts to solve a real-world problem.

Exchange Rate Function

An exchange rate function, such as \( f(x) = 0.82x \), models how one currency is converted into another. In this example, for every 1 U.S. dollar, you get 0.82 euros. This conversion factor is the heart of the exchange rate function. It is important to understand that such functions can fluctuate over time based on economic factors, but this exercise assumes a static rate for simplicity.

To elaborate, if you have 100 U.S. dollars, the function \( f(100) \) tells you its equivalent in euros. Exchange rate functions are essential for businesses operating internationally and travelers alike. They are also an excellent example of how a linear function can model real-world relationships between variables—in this instance, the amount in one currency and its value in another.

Revenue Function

The revenue function \( R(t) = 40 + 2t \) describes how a company's revenue changes over time. In this context, R(t) is the revenue in millions of U.S. dollars and t represents the years since 2003. The constant term '40' indicates the base revenue at the starting point, which is the year 2003, and the coefficient '2' shows the rate at which the revenue increases every year.

Revenue functions are indispensable in business for projecting earnings and making informed decisions about investments, growth, and strategizing for future actions. The linearity of this particular revenue function suggests a steady, constant growth over time, a situation often desired for financial stability and long-term planning. Additionally, when a revenue function is combined with an exchange rate function through composition, the resulting function provides international insight by translating the revenue into a foreign currency, which can be essential for global economic analysis.