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Question 1 12 pts Let A = {1, 2, 3, 4}. Let the functions F, G, and H be given with domain and codomain A defined as F(1)=3,F(2)=2,F(3)=2,and F(4)=4; G(1) =1,G(2) =3,G(3) = 4, and G(4) = 2; H(1) =2, H(2) =4, H(3) = 1, and H(4) = 3. Find the following: (a) Fo G (b) Ho F (c) G o H (d) Fo Go H Edit View Insert Format Tools Table 12pt v Paragraph V B I U AV & VTV V 3º V V V : Time Attem 5 Da' Secol p 0 words Question 2 10 pts Given: f ={(a, 1), (b, 2), (c, 3), (d, 4)} g-1={(2,1),(3,2),(4,3), (1,4)} Verify the following: (g o f)-1 (x) = f-1 og-1(x) Edit View Insert Format Tools Table 12pt v V Paragraph BIUAV T2 > V V D V V : p 0 words > Question 3 10 pts If f : R\{-1} -> R\{1} is defined by f(x) = 1+x 1-x and g : R -> R is defined by g(x) = 1+22, then 2x prove that f o g(x) =(f(x))2. Edit View Insert Format Tools Table 12pt v Paragraph B I U A V & V TV 00 V > > ... p 0 words Question 4 8 pts Let f(x) = ax + b and g(x) = cx + d, where a, b, c, and d are constants. Determine necessary and sufficient conditions on the constants a, b, c, and d so that f og = go f. Edit View Insert Format Tools Table 12pt v V Paragraph B I U A V & V TV V V A VE V : p 0 words Question 5 9 pts Prove the following using the Pigeonhole Principle. (a) Prove that among any group of 27 English words, there are at least two words that start with the same letter. (b) Prove that among any group of 13 people, there are at least two people who have the same number of friends within the group, when each person does not know atleast one person. (c) In a group of 101 people, each person in the group knows a unique set of 10 languages. Prove that there are two people in the group who can communicate with each other in at least two different languages. Edit View Insert Format Tools Table 12pt Paragraph v B I U A V & V TV 20 V V D V : p 0 words 7 Question 6 16 pts For the following sequences find their closed AND recursive expressions. Be sure to add in all parts for both (a) (3, 7, 11, 15, ... ) (b) (2, 6, 12, 20, ... ) (c) (1, -2, 3, -4, ... ) (d) (1, 3, 6, 10, ... ) Edit View Insert Format Tools Table 12pt Paragraph V B I U A V & V TV 00 V V V V ... p 0 words 1 Question 7 12 pts Use and/or ][ notation to rewrite the following (a) (x+y+1)-(x+y+2)+(x+y+3) -... +(x+y+57) (b) (x+1)(x2 +2)(x3+3) ... (x20 +20) (c) 1(0+1) +2(0+1+2)+3(0+1+2+3) Edit View Insert Format Tools Table 12pt v V Paragraph BIUA > > T2 V 30 V V A > ilılı V : p 0 words

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