Q. 73 - Precalculus Enhanced with Graphing Utilities

Question

Foreign Exchange Traders often buy foreign currency in hope of making money when the currency’s value changes.

For example, on March 22, 2011, one U.S. dollar could purchase 0.7045 Euros, and one Euro could purchase 114.9278 yen. Let $f\left(x\right)$ represent the number of Euros you can buy with $x$ dollars, and let $g\left(x\right)$ 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 $\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)$

(d) What is $g\left(f\left(1000\right)\right)$? Interpret this result.

Step-by-Step Solution

Answer

(a) $f\left(x\right)=0.7045x$

(b)$g\left(x\right)=114.9278x$

(c) $\left(g\circ f\right)\left(x\right)=80.9666351x$

(d) $80966.64$

Step 1. Given Information

On March 22, 2011, one U.S. dollar could purchase 0.7045 Euros, and one Euro could purchase $114.9278$ yen. Let $f\left(x\right)$ represent the number of Euros you can buy with $x$ dollars, and let $g\left(x\right)$represent the number of yen you can buy with $x$Euros.

Part (a) Step 1. Finding function that relates dollars to Euros.

Let $x=$amount of dollars

Let $f\left(x\right)$ represent the number of Euros you can buy with $x$ dollars,

The given states that one US dollar can buy $0.7045$dollars.

Thus, the function that relates dollars to Euros is $f\left(x\right)=0.7045x$

Part (b) Step 1. Function that relates Euros to yen.

Let $x=$amount in Euros

Let $g\left(x\right)$ represent the number of Euros you can buy with style="width:30%" $x$ dollars,

The given states that one Euro can buy $114.9278$Yen.

Thus, $g\left(x\right)=114.9278x$

Part (c) Step 1. Function that relates dollars to yen

The function that relates dollars to yen

 $$\begin{gathered} \left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)=114.9278\left(0.7045x\right) \\ =80.9666351x \end{gathered}$$

Part (d) Finding g f 1000

$$\begin{gathered} g\left(f\left(1000\right)\right)=80.9666351\left(1000\right) \\ g\left(f\left(1000\right)\right)=80966.64 \end{gathered}$$