Question

type the correct answer in each box. use numerals instead of words. if necessary, round to the nearest hundredth.
the relationship between the values of two different currencies is modeled by a linear function that varies depending on the current value of each currency. the table shows the relationship between an amount in u.s. dollars, x, and the equivalent amount in euros, f(x), where f(x) = \frac{x}{1.06}.

x	1	1.06	2	2.12
f(x)	0.95	1	1.89	2

the inverse function, f⁻¹, is given by f⁻¹(x) = \boxed{} x.
2 euros are equivalent to \boxed{} u.s. dollars.
4 euros are equivalent to \boxed{} u.s. dollars.

Explanation:

Step1: Find the inverse function

Given \( f(x)=\frac{x}{1.06} \), let \( y = f(x) \), so \( y=\frac{x}{1.06} \). To find the inverse, swap \( x \) and \( y \) and solve for \( y \):
\( x=\frac{y}{1.06} \), multiply both sides by \( 1.06 \): \( y = 1.06x \). Thus, \( f^{-1}(x)=1.06x \).

Step2: 2 euros to dollars

Use \( f^{-1}(x) \) with \( x = 2 \): \( f^{-1}(2)=1.06\times2 = 2.12 \).

Step3: 4 euros to dollars

Use \( f^{-1}(x) \) with \( x = 4 \): \( f^{-1}(4)=1.06\times4 = 4.24 \).

Answer:

For the inverse function: \( 1.06 \)
For 2 euros: \( 2.12 \)
For 4 euros: \( 4.24 \)