Define the following functions f, g, and h: f: {0, 1}3{0, 1}3. The output of f is obtained by taking the input string and replacing the first bit by 1, regardless of whether the first bit is a 0 or 1. For example, f(001) = 101 and f(110) = 110. g: {0, 1}3{0, 1}3. The output of g is obtained by taking the input string and reversing the bits. For example, g(011) = 110. h: {0, 1}3{0, 1}3. The output of h is obtained by taking the input string x, and replacing the last bit with a copy of the first bit. For example, h(011) = 010. (a) What is g f(010)? (b) What is g h(010)? (c) What is h f(010)? (d) What is the range of h f? (e) What is the range of g f? (a) g o f (0 1 0) f (0 1 0) = 110 g o f (0 1 0) = g [ f (010)] g [f (010)] =g (110) = 011 (b) g o h (0 1 0) h ( 0 1 0) = 010 g (010) = 010 c) h o (f 010) h (110) =111 d) h o f =h[f] f [(100), (101), (110), (111)] h o f = [(101), (111)] Problem 5 Given functions F(x) = x2 g (x) = 2x h(x) = x/5 a) Fog(0) = f [g(0)] = f(20) =f(1) = 12 = 1 b) Foh (52) = f[h (s2) ] =f(52/5) = (52/5) 2 =2704/25 c) g o h o f(4) = g o h [f(4)] = goh(42) =goh(16) =g [h(16)] =g(16/5) =216/5 d) hof =h[f(x)] = h(x2) = x2/5 e) fog =f g(x) = f(2x) =(2x)2 Define the following functions f, g, and h: 3 3 f : {0, 1 } { 0, 1 } . The output of f is obtained by taking the input string and replacing the first bit by 1, regardless of whether the first bit is a 0 or 1. For example, f(001) = 101 and f(110) = 110. 3 g : { 0,1 } {0, 1 } 3 . The output of g is obtained by taking the input string and reversing the bits. For example, g(011) = 110. 3 3 h : { 0, 1 } { 0,1 } . The output of h is obtained by taking the input string x, and replacing the last bit with a copy of the first bit. For example, h(011) = 010. (a) What is g f(010)? (b) What is g h(010)? (c) What is h f(010)? (d) What is the range of h f? (e) What is the range of g f? (a) f (010) ( g o f )(0 1 0)=g g (110) 011 (b) [using the definition of g, g(110) = 011] h(010) ( g o h)(0 1 0)=g g (0 10) 010 (c) [using the definition of f, f(010) = 110] [using the definition of h, h(010) = 010] [using the definition of g, g(010) = 010] f (010) (h o f )(01 0)=h h(1 10) 111 (f) To find the range of [using the definition of f, f(010) = 110] [using the definition of h, h(010) =111] h o f , we shall find the domain of ho f the domain of f . Domain of f = {(000),(001),(010),(100),(011),(101),(110),(111)} Domain of h in h o f is the range of f = first, which is also {f(000),f(001),f(010),f(100),f(011),f(101),f(110),f(111)} = {(100),(101),(110),(111)} Range of h o f = {h(100),h(101),h(110),h(111)} = {101,111} (e) To find the range of g o f , we shall find the domain of gof first, which is also the domain of f . Domain of f = {(000),(001),(010),(100),(011),(101),(110),(111)} Domain of g in g o f is the range of f = {f(000),f(001),f(010),f(100),f(011),f(101),f(110),f(111)} = {(100),(101),(110),(111)} Range of g o f = {g(100),g(101),g(110),g(111)} = {001,101,011,111} Problem 5 Given functions f(x) = x2 , g (x) = 2x , h(x) = x/5 a) (fog)(0)=f (g(0))=f (20 ) 12 [since f(1) = [since g(0) = 20] f (1) 12] 1 b) (foh)(52)=f (h(52))=f 52 5 ( ) [since h(52) = 52/5] 2704 12] 25 c) ( g o h o f )(4)=( g o h)( f ( 4 ) ) ( goh)(4 2) ( goh) ( 16 ) g (h ( 16 ) ) [since f(4) = 42] 52 5 2 ( ) [since f(1) = g ( 165 ) 16 = 25 d) (hof )( x)=h( f ( x ) ) h(x 2) 2 e) x 5 (fog )( x)=f (g ( x )) f (2x ) 22 x [since h(16) = 16/5] 16 [since g(16/5) = 2 5 ]