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Inverse Functions and Compositions of Functions






Q. Let f be the function from {a, b, c} to {1, 2, 3} such that f(a) = 2, f(b) = 3, and f(c) = 1. Is f invertible, and if it is, what is its inverse?



Q. Let f : Z → Z be such that f(x) = x + 1. Is f invertible, and if it is, what is its inverse?





Composition of the functions 'f' and 'g'





Q. Let g be the function from the set {a, b, c} to itself such that g(a) = b, g(b) = c, and g(c) = a. Let f be the function from the set {a, b, c} to the set {1, 2, 3} such that f(a) = 3, f(b) = 2, and f(c) = 1. What is the composition of f and g, and what is the composition of g and f ?





Q. Let f and g be the functions from the set of integers to the set of integers defined by f(x) = 2x + 3 and g(x) = 3x + 2. What is the composition of f and g? What is the composition of g and f ?






The Graphs of Functions




Q. Display the graph of the function f (n) = 2n + 1 from the set of integers to the set of integers.



Q. Display the graph of the function f(x) = x2 from the set of integers to the set of integers.





Ceiling and Floor functions :




Useful Properties of the Floor and Ceiling Functions





Property of ceil Property of floor
⌈x⌉ = n if and only if n − 1 < x ≤ n ⌊x⌋ = n if and only if n ≤ x < n+ 1
⌈x⌉ = n if and only if x ≤ n < x + 1 ⌊x⌋ = n if and only if x − 1 < n ≤ x
x − 1 < ⌊x⌋ ≤ x ≤ ⌈x⌉ < x+ 1
⌈-x⌉ = −⌈x⌉ ⌊-x⌋ = − ⌊x⌋
⌈x + n⌉ = ⌈x⌉ + n ⌊x + n⌋ = ⌊-x⌋ + n




Sequences and Summations




The sequences {sn} with sn = −1 + 4n and {tsn} with tsn = 7 − 3n are both arithmetic progressions with initial terms and common differences equal to −1 and 4, and 7 and −3, respectively, if we start at n = 0. The list of terms s1, s2, s3 ... sn



Recurrence Relations



Q. Let {an} be a sequence that satisfies the recurrence relation an = an-1 + 3 for n = 1, 2, 3,..., and suppose that a0 = 2. What are a1, a2, and a3?



Q. The Fibonacci sequence, f0 = 0, f1 = 1 and the recurrence relation fn = fn-1 + fn-2 for n = 2,3...



Important Summations






Cardinality of Sets




Countable Sets



Q. Show that the set of odd positive integers is a countable set.



Q. Show that the set of all integers is countable.



Useful Theorems




Matrices




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