IM Alg1.4.15 Lesson: Inverse Functions
Here is an encoded message, a message that has been converted into a code. WRGDB LV D JRRG GDB. Can you figure out what it says in English? How was the original message encoded?
How was the original message encoded?
Now it’s your turn to write a secret code!
Write a short and friendly message with 3–4 words.
Pick a number from 1 to 10. Then, encode your message by shifting each letter that many steps forward or backward in the alphabet, wrapping around from Z to A as needed. Consider using the table below to create a key for your cipher. Then, give your encoded message to a partner to decode. If requested, give the number you used. Decode the message from your partner. Ask for their number, if needed.
Suppose m and c each represent the position number of a letter in the alphabet, but m represents the letters in the original message and c the letters in your secret code. Complete the table.
Use and to write an equation that can be used to encode an original message into your secret code.
Use and to write an equation that can be used to decode your secret code into the original message.
There are 26 letters in the alphabet, so only the numbers 1–26 make sense for m and c.
Try using the equation that you wrote to encode the letters A, B, Y, and Z. Did you end up with position numbers or values that are less than 1 or greater than 26? For which letters?
Use your encoding equation to plot the (m,c) pairs for all the letters in the alphabet.
Look for the points whose value is less than 1 or greater than 26. What letters should they be in the code? Plot the points where they should be according to the rule of your cipher.
Did you end up with a graph of a piecewise function? If so, can you describe the different rules that apply to different domains of the function?
An American traveler who is heading to Mexico exchanges some U.S. dollars for Mexican pesos. At the time of his travel, 1 dollar can be exchanged for 19.32 pesos.
Find the amount of money in pesos that the American traveler would get if he exchanged 100 dollars.At the same time, a Mexican businesswoman who is in the United States is exchanging some Mexican pesos for U.S. dollars at the same exchange rate. 
Find the amount of money in pesos that the American traveler would get if he exchanged 500 dollars.
Write an equation that gives the amount of money in pesos, , as a function of the dollar amount, , being exchanged.
Find the amount that the Mexican businesswoman would get if she exchanged 1,000 pesos.
Find the amount that the Mexican businesswoman would get if she exchanged 5,000 pesos.
Explain why it might be helpful to write the inverse of the function you wrote earlier. Then, write an equation that defines the inverse function.