91Ó°ÊÓ

When working with currency conversions, inverse functions play a crucial role. An inverse function essentially 'reverses' another function. If you have a function that converts dollars to euros, the inverse will convert euros back to dollars, and vice versa.
To show that two functions are inverses of each other, you must demonstrate that applying one function and then the other returns the original value. Mathematically, this means, for functions \( f \) and \( g \), they are inverses if:

• \f{ f(g(x)) = x }
• \f{ g(f(x)) = x }

In the exercise, the conversion from dollars to euros is given by \( C_1(d) = 0.749d \). Its inverse, converting euros back to dollars, is \( C_2(r) = \frac{r}{0.749} \).
By evaluating:

• \f{ C_2(C_1(d)) = C_2(0.749d) = \frac{0.749d}{0.749} = d }
• \f{ C_1(C_2(r)) = C_1\left(\frac{r}{0.749}\right) = 0.749 \times \frac{r}{0.749} = r }

These calculations confirm that \( C_1 \) and \( C_2 \) are indeed inverse functions.

Exchange rates indicate how much of one currency you can exchange for another. It's a crucial concept in international finance, travel, and trade. In the problem, the exchange rate from U.S. dollars to euros is given as 0.749. This means you can trade 1 U.S. dollar for 0.749 euros.
Exchange rates fluctuate due to various factors:

• Market demand and supply
• Economic indicators
• Political stability
• Interest rates

Understanding these shifts helps in making informed decisions about currency conversions. For example, if the exchange rate changes and 1 U.S. dollar equals 0.8 euros, the value of the U.S. dollar has increased relative to the euro.
Using the current exchange rate in our example:

• \( C_1(d) = 0.749d \) converts dollars to euros.
• \( C_2(r) = \frac{r}{0.749} \) converts euros back to dollars.

Knowing the exchange rate allows you to construct these conversion functions and use them for practical computations.

Function evaluation is the process of substituting a specific value for the variable in a function and then performing the operations defined by the function. This is a fundamental concept in mathematics that applies to various real-world scenarios, including currency conversion.
For example, given the function \( C_1(d) = 0.749d \), to find out how many euros 25 dollars would convert to, you substitute 25 for \(d\):
\( C_1(25) = 0.749 \times 25 = 18.725 \text{ euros} \)
Similarly, for the inverse function \( C_2(r) = \frac{r}{0.749} \), to find out how many dollars 100 euros would convert to, you substitute 100 for \( r \):
\( C_2(100) = \frac{100}{0.749} \approx 133.51 \text{ dollars} \)
By evaluating these functions at specific values, you can solve real-life problems easily. Function evaluation provides clarity and precision in applying mathematical functions to everyday tasks, such as financial planning and currency exchange.