[T] An American tourist visits Paris and must convert U.S....

[T] An American tourist visits Paris and must convert U.S. dollars to Euros, which can be done using the function $E(x)=0.79 x,$ where $x$ is the number of U.S. dollars and $E(x)$ is the equivalent number of Euros. Since conversion rates fluctuate, when the tourist returns to the United States 2 weeks later, the conversion from Euros to U.S. dollars is $D(x)=1.245 x,$ where $x$ is the number of Euros and $D(x)$ is the equivalent number of U.S. dollars. a. Find the composite function that converts directly from U.S. dollars to U.S. dollars via Euros. Did this tourist lose value in the conversion process? b. Use (a) to determine how many U.S. dollars the tourist would get back at the end of her trip if she converted an extra $\$ 200$ when she arrived in Paris.

[T] An American tourist visits Paris and must convert U.S. dollars to Euros, which can be done using the function $E(x)=0.79 x,$ where $x$ is the number of U.S. dollars and $E(x)$ is the equivalent number of Euros. Since conversion rates fluctuate, when the tourist returns to the United States 2 weeks later, the conversion from Euros to U.S. dollars is $D(x)=1.245 x,$ where $x$ is the number of Euros and
$D(x)$ is the equivalent number of U.S. dollars.
a. Find the composite function that converts directly from U.S. dollars to U.S. dollars via Euros. Did this tourist lose value in the conversion process?
b. Use (a) to determine how many U.S. dollars the tourist would get back at the end of her trip if she
converted an extra $\$ 200$ when she arrived in Paris.

Key Concepts

Analysis of Net Value Change

Analyzing net value change involves comparing the initial amount before conversion to the final amount after performing a composite of conversion functions. This concept helps determine whether any loss or gain has occurred during the process, which is an important aspect in transaction and finance modeling.

Composite Functions

Composite functions involve applying one function to the result of another. In mathematics, this allows two or more actions to be combined into a single operation. This concept is crucial when modeling multi-step processes such as converting currencies and then converting them back, as it shows how the combined effect of both functions can be determined.

Linear Functions

Linear functions describe relationships with a constant rate of change, typically expressed in the form f(x) = mx + b. In the context of currency conversion, these functions demonstrate how an input value is directly scaled by a conversion factor, simplifying the process of understanding proportional relationships.

Currency Exchange and Conversion Rates

Currency exchange uses conversion rates as multiplicative factors to change monetary values from one unit to another. This concept models real-world situations where linear functions are applied to determine equivalence between different currencies, enabling the analysis of value changes in transactions.

Transcript

00:01 Hey everyone, today we're solving problem number 56, which is a word problem from chapter 1, section 1 of the textbook.

00:10 Essentially, the situation given is that an american tourist visits paris and must convert us dollars to euros, which can be done using the function e of x equals 0 .79x, so that's one of them.

00:40 And this is where x is the number of us dollars and e is the equivalent number of euros.

00:52 So this is converting dollars to euros.

01:03 I believe euros is like a roundy, something like that.

01:08 And then we're also told later on in the problem.

01:12 Since conversion rates fluctuate when the tourist returns to the united states two weeks later, the conversion from euros to united states dollars becomes d of x equals 1 .245x.

01:37 And again, this is for the american tourists to convert his or her.

01:42 Euros now back into american dollars after his or her trip to.

01:54 So we're asked to essentially in part a write the composite function that converts directly from us dollars to us dollars via euros.

02:05 So essentially what this is asking for is d of e of x, which is written like that or you can write it as d of in parentheses e of x the same thing that's our composite function and the way you're going to do this is that you know that you have 0 .79 times x and 1 .245 times x so i'm going to essentially have 0 .79 nope sorry 1 .245 of 0 .79 x so when i multiply out 1 .245 by 0 .79 my composite function will be 0 .79.

03:18 9 ,8, 355 times x, which if we notice is less than one.

03:29 So that means for every 98 us cents, she's losing money, essentially.

03:39 So that answers the second part of the first part, the question posed in the first part, which is, did this tourist lose value in the conversion process? yes, because she's only making back every 98 cents of the initial dollar she left with...

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