Composition of Functions ACTIVITY Code Breakers 5 - …€¦ · · 2014-05-28Composition of Functions Code Breakers ... types of functions also have inverses. ... Graph f and f-1
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5.1Composition of FunctionsCode BreakersSUGGESTED LEARNING STRATEGIES: Shared Reading, Marking the Text, Summarize/Paraphrase/Retell, Look for a Pattern, Quickwrite
Th e use of cryptography goes back to ancient times. In ancient Greece, Spartan generals exchanged messages by wrapping them around a rod called a scytale and writing a message on the adjoining edges. Th e Roman general and statesman Julius Caesar used a transposition cipher that translated letters three places forward in the alphabet. For example, the word CAT was encoded as FDW.
In modern times, cryptography was used to secure electronic communications. Soon aft er Samuel F.B. Morse invented the telegraph in 1844, its users began to encode the messages with a secret code, so that only the intended recipient could decode them. During World War II, British and Polish cryptanalysts used computers to break the German Enigma code so that secret messages could be deciphered.
Many young children practice a form of cryptography when writing notes in secret codes. Th e message below is written in a secret code.
1. Try to decipher the seven-letter word coded above.
2. What do you need to decipher the seven-letter word?
CONNECT TO COMPUTINGCOMPUTING
Modern computers have completely changed the science of cryptography. Before computers, cryptography was limited to two basic types: transposition, rearranging the letters in a message, and substitution, replacing one letter with another. The most sophisticated pre-computer codes used fi ve or six operations. Computers can now use thousands of complex algebraic operations to encrypt messages.
CONNECT TO SCIENCESCIENCE
Cryptography is the science of code-making (encoding) and cryptanalysis is the science of code-breaking (decoding).
268 SpringBoard® Mathematics with MeaningTM Algebra 2
My Notes
Composition of FunctionsACTIVITY 5.1continued Code BreakersCode Breakers
SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Create Representations, Marking the Text, Summarize/Paraphrase/Retell
3. Th e following message uses a numerical code. Can you decode the four-letter word? Explain how you know.
13 1 20 8
4. What is this six-letter word?
21 5 10 17 17 14
In Item 3, a single function was used to encode a word. Th e function assigned each letter to the number representing its position in the alphabet.
In Item 4, two functions were used to encode a word. Th e fi rst function assigned each letter to the number representing its position in the alphabet x and then the function f(x) = x + 2 was used to encode the message further as shown in the table.
LETTER x f(x)S 19 21C 3 5H 8 10O 15 17O 15 17L 12 14
LETTER-TO-NUMBER CODES
A = 1 N = 14B = 2 O = 15C = 3 P = 16D = 4 Q = 17E = 5 R = 18F = 6 S = 19G = 7 T = 20H = 8 U = 21I = 9 V = 22
J = 10 W = 23K = 11 X = 24L = 12 Y = 25M = 13 Z = 26
Composition of Functions Code BreakersCode Breakers
SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Work Backward
5. Write a function g that could decode the message in Item 4 and use it to complete the table below.
g(x) =
x g(x) LETTER21 S5 C
10 H17 O17 O14 L
6. Try to decipher the more diffi cult message below. First, each letter in the message was assigned a number based on its position in the alphabet, and then another function encoded the message further.
20 -1 50 8 11 50
7. Th e encoding function for Item 6 is f(x) = .
Write a decoding function g and complete the table below.
270 SpringBoard® Mathematics with MeaningTM Algebra 2
My Notes
Composition of FunctionsACTIVITY 5.1continued Code BreakersCode Breakers
SUGGESTED LEARNING STRATEGIES: Guess and Check, Interactive Word Wall, Vocabulary Organizer, Create Representations, Simplify the Problem, Think/Pair/Share
8. Verify that your function works by encoding the letter C = 3 with f and then decoding it by using g.
Recall that functions f and g are called inverse functions if and only if f(g(x)) = x for all x in the domain of g and g( f(x)) = x for all x in the domain of f.
9. Use the defi nition of inverse functions to show that the encoding function f(x) = 3x - 4 and its decoding function g are inverses.
10. What is f -1 for the function f(x) = x + 2?
11. What is f -1 for the function f(x) = 3x - 4?
READING MATH
The symbol for the inverse of function f is f -1 . You read this symbol as “f inverse.”
f( g(x)) indicates a composition of functions, in which the range of function g becomes the domain for function f.
CONNECT TO APAP
The notion of inverse is a very important concept in the study of functions in advanced mathematics courses.
A function is defi ned as one-to-one if, for each number in the range of the function, there is exactly one corresponding number in the domain of the function.
14. Is g from Item 13 a one-to-one function? Explain.
15. What do you know about a function whose inverse relation is a function?
16. Determine whether each type of function will always have an inverse that is a function. Explain your reasoning.
a. Linear function
b. Quadratic function
17. Investigate the cubic function h(x) = x 3 - 6 x 2 + 8x + 5.
a. Use your calculator to graph h in the viewing window [-10, 16] by [-10, 16] and sketch the results in the My Notes section.
b. Explain whether or not h is a one-to-one function.
ACADEMIC VOCABULARY
A one-to-one function is a function in which for each element of the range, there is exactly one corresponding element in the domain.
274 SpringBoard® Mathematics with MeaningTM Algebra 2
My Notes
Composition of FunctionsACTIVITY 5.1continued Code BreakersCode Breakers
SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Create Representations, Quickwrite
17. (continued) c. Is your answer in Part (b) true for all cubic functions? Explain.
d. Does a cubic function always have an inverse that is a function? Explain your reasoning.
Th ree students are encoding and decoding messages that begin with the numerical code used in Item 3. Suppose that the message HELLO is to be passed from one student to a second student and then on to a third student. Use this information for Items 18–24.
18. Th e fi rst student translates HELLO to numbers and then encodes it with the function f(x) = -2x + 12. What encoded message will the second student receive?
19. Aft er receiving the encoded message, the second student encodes it again, using the function g(x) = -x + 9. What encoded message will the third student receive?
20. Let h be the composite function h(x) = g( f(x)).
a. Write a rule for h.
b. Explain how h relates to the encoding of the message HELLO.
Composition of Functions Code BreakersCode Breakers
SUGGESTED LEARNING STRATEGIES: Work Backward, Create Representations, Quickwrite, Summarize/Paraphrase/Retell, Look for a Pattern
21. Th e third student receives another message shown below. What does this message say?
27 25 7 37 27 27 25 7
22. Suppose that the message above had been encoded using the composite function k(x) = f ( g(x)). How would the message have been encoded?
23. Can the rule found in Item 20(a) also be used for k? Explain.
24. Suppose that the third student is given f and g, and the encoded message. What additional information would help this student to decode the message more effi ciently? Explain.
25. Another message is encoded twice. First, the message is encoded with the function f(x) = 2x - 29, and then it is encoded with the function g(x) = x 2 .
a. Encode the word GRAPH using the composite function.