functions and their inverses can be used to encode and decode messages. To encode and decode a message, first replace each letter of the alphabet with a positive integer using the following scheme, thus rewriting the original message as numbers instead of words: P - 16 Q-17 R- 18 S-19 T-20 А — 1 В - 2 С -3 D-4 E-5 K-11 L-12 М- 13 N- 14 O-15 U- 21 V-22 W- 23 X-24 Y-25 Z-26 F-6 G-7 Blank - 27 H-8 T-9 J-10 ENCODING: A one-to-one function can be used to encode a numerical message. For example, suppose you want to send the message MATH to a friend, and you have decided that the function f(x) = 3x + 4 will be the encoding function. This function simply describes the procedure used to create the encoded message – in this case multiply by 3 and add 4. First change the letters to corresponding numbers as shown above: 13 1 20 8. Then use these as the input values in f(x). en won uoy Jud nolton f(13) = 3(13) + 4 = 43 f(1) = 7 f(20) = 64 f(8) = 28 noitee pniwolict oN rb gnia how auolveg mon eonil ol beau ed bioo ier So the encoded message that you send to your friend is: 43 7 64 28 ens NOTE: A graphing calculator can be used to evaluate a function as above: a. Press [Y=Lon the calculator. Type 3x + 4. b. Choose [2nd[WINDOW] (TBLSET). Set the independent variable to Indpnt: Ask. c. Then choose [2nd][GRAPH] (TABLE). Input each x value (13 1 20 8). The calculator will return the function values (43 7 64 28) which is the encoded message. DECODING an encoded message: Now it is up to your friend to decode the message. Decodin is the process that "undoes" the encoding process. If f(x) encodes the message, what will decoc it? The inverse of f(x) or f'(x)! So f'(x) will be the decoding function. X-4 In the example, the inverse of f(x) = 3x + 4 can be shown to be f'(x) Take the encode 3 %3D message (43 7 64 28) and use these values as input values in f'(x). Again, the calculator ca X-4 be used to decode - simply enter y = and use the TABLE feature as described above. f'(43) = 13 f'(7) = 1 f'(64) = 20 f'(28) = 8 That's the original numerical message! The last step is to convert back to letters using the ta given previously. Now your friend knows the message that you sent: MATH.