Please Tell Me in Dollars and Cents Learning Task
1. Aisha made a chart of the experimental data for her science project and showed it to her science teacher. The teacher was complimentary of Aisha’s work but suggested that, for a science project, it would be better to list the temperature data in degrees Celsius rather than degrees Fahrenheit.
a. Aisha found the formula for converting from degrees Fahrenheit to degrees Celsius: .
Use this formula to convert freezing (32°F) and boiling (212°F) to degrees Celsius.
b. Later Aisha found a scientific journal article related to her project and planned to use information from the article on her poster for the school science fair. The article included temperature data in degrees Kelvin. Aisha talked to her science teacher again, and they concluded that she should convert her temperature data again – this time to degrees Kelvin. The formula for converting degrees Celsius to degrees Kelvin is
.
Use this formula and the results of part a to express freezing and boiling in degrees Kelvin.
c. Use the formulas from part a and part b to convert the following to °K: – 238°F, 5000°F .
In converting from degrees Fahrenheit to degrees Kelvin, you used two functions, the function for converting from degrees Fahrenheit to degrees Celsius and the function for converting from degrees Celsius to degrees Kelvin, and a procedure that is the key idea in an operation on functions called composition of functions.
Composition of functions is defined as follows: If f and g are functions, the composite function (read this notation as “f composed with g) is the function with the formula
,
where x is in the domain of g and g(x) is in the domain of f.
2. We now explore how the temperature conversions from Item 1, part c, provide an example of a composite function.
a. The definition of composition of functions indicates that we start with a value, x, and first use this value as input to the function g. In our temperature conversion, we started with a temperature in degrees Fahrenheit and used the formula to convert to degrees Celsius, so the function g should convert from Fahrenheit to Celsius: . What is the meaning of x and what is the meaning of g(x) when we use this notation?
b. In converting temperature from degrees Fahrenheit to degrees Kelvin, the second step is converting a Celsius temperature to a Kelvin temperature. The function f should give us this conversion; thus,. What is the meaning of x and what is the meaning of f (x) when we use this notation?
c. Calculate . What is the meaning of this number?
d. Calculate , and simplify the result. What is the meaning of x and what is the meaning of ?
e. Calculate using the formula from part d. Does your answer agree with your calculation from part c?
f. Calculate , and simplify the result. What is the meaning of x? What meaning, if any, relative to temperature conversion can be associated with the value of ?
We now explore function composition further using the context of converting from one type of currency to another.
3. On the afternoon of May 3, 2009, each Japanese yen (JPY) was worth 0.138616 Mexican pesos (MXN), each Mexican peso was worth 0.0547265 Euro (EUR), and each Euro was worth 1.32615 US dollars (USD).[1]
a. Using the rates above, write a function P such that P(x) is the number of Mexican pesos equivalent to x Japanese yen.
b. Using the rates above, write a function E that converts from Mexican pesos to Euros.
c. Using the rates above, write a function D that converts from Euros to US dollars.
d. Using functions as needed from parts a – c above, what is the name of the composite function that converts Japanese yen to Euros? Find a formula for this function. (Original values have six significant digits; use six significant digits in the answer.)
e. Using functions as needed from parts a – c above, what is the name of the composite function that converts Mexican pesos to US dollars? Find a formula for this function. (Use six significant digits in the answer.)
f. Using functions as needed from parts a – c above, what is the name of the composite function that converts Japanese yen to US dollars? Find a formula for this function. (Use six significant digits in the answer.)
g. Use the appropriate function(s) from parts a - f to find the value, in US dollars, of the following: 10,000 Japanese yen; 10,000 Mexican pesos; 10,000 Euros.
Returning to the story of Aisha and her science project: it turned out that Aisha’s project was selected to compete at the science fair for the school district. However, the judges made one suggestion – that Aisha express temperatures in degrees Celsius rather than degrees Kelvin. For her project data, Aisha just returned to the values she had calculated when she first converted from Fahrenheit to Celsius. However, she still needed to convert the temperatures in the scientific journal article from Kevin to Celsius. The next item explores the formula for converting from Kelvin back to Celsius.
4. Remember that the formula for converting from degrees Celsius to degrees Kelvin is
.
In Item 2, part b, we wrote this same formula by using the function f where represents the Kelvin temperature corresponding to a temperature of x degrees Celsius.
a. Find a formula for C in terms of K, that is, give a conversion formula for going from °K to °C.
b. Write a function h such that is the Celsius temperature corresponding to a temperature of x degrees Kelvin.
c. Explain in words the process for converting from degrees Celsius to degrees Kelvin. Do the equation and the function f from Item 2, part b both express this idea?
d. Explain verbally the process for converting form degrees Kelvin to degrees Celsius. Do your formula from part a above and your function h from part b both express this idea?
e. Calculate the composite function , and simplify your answer. What is the meaning of x when we use x as input to this function?
f. Calculate the composite function , and simplify your answer. What is the meaning of x when we use x as input to this function?
In working with the functions f and h in Item 4, when we start with an input number, apply one function, and then use the output from the first function as the input to the other function, the final output is the starting input number. Your calculations of and show that this happens for any choice for the number x. Because of this special relationship between f and h , the function h is called the inverse of the function f and we use the notation (read this as “f inverse”) as another name for the function h.
The precise definition for inverse functions is: If f and h are two functions such that
for each input x in the domain of f,
and
for each input x in the domain of h,
then h is the inverse of the function f, and we write h = . Also, f is the inverse of the function h, and we can write f = .
Note that the notation for inverse functions looks like the notation for reciprocals, but in the inverse function notation, the exponent of “–1 ” does not indicate a reciprocal.
5. Each of the following describes the action of a function f on any real number input. For each part, describe in words the action of the inverse function, , on any real number input. Remember that the composite action of the two functions should get us back to the original input.
a. Action of the function f : subtract ten from each input
Action of the function :
b. Action of the function f : add two-thirds to each input
Action of the function :
c. Action of the function f : multiply each input by one-half
Action of the function :
d. Action of the function f : multiply each input by three-fifths and add eight
Action of the function :
6. For each part of Item 5 above, write an algebraic rule for the function and then verify that the rules give the correct inverse relationship by showing that and for any real number x.
Before proceeding any further, we need to point out that there are many functions that do not have an inverse function. We’ll learn how to test functions to see if they have an inverse in the next task. The remainder of this task focuses on functions that have inverses. A function that has an inverse function is called invertible.
7. The tables below give selected values for a function f and its inverse function .
a. Use the given values and the definition of inverse function to complete both tables.
x3
5 / 10
7 / 6
3
11 / 1
x / f (x)
11
3 / 9
7
10
15 / 3
x
3 / 15
5 / 10
7 / 6
9 / 3
11 / 1
x / f (x)
1 / 11
3 / 9
6 / 7
10 / 5
15 / 3
b. For any point (a, b) on the graph of f, what is the corresponding point on the graph of ?
c. For any point (b, a) on the graph of , what is the corresponding point on the graph of f ? Justify your answer.
As you have seen in working through Item 7, if f is an invertible function f and a is the input for function f that gives b as output, then b is the input to the function that gives a as output. Conversely, if f is an invertible function f and b is the input to the function that gives a as output, then a is the input for function f that gives b as output.
[1] Students may find it more interesting to look up current exchange values to use for this item and Item 9, which depends on it. There are many websites that provide rates of exchange for currency. Note that these rates change many times throughout the day, so it is impossible to do calculations with truly “current” exchange values. The values in Item 3 were found using http://www.xe.com/ucc/ .