Question

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8. All compositions are from right to left (a) What is meant when a function f is said to be: surjective; injective; a bijection. When is a function invertible? [4 pt.] (b) Consider the function f : R ? R given by f(z) = (1-x)2 i. Write down the domain and range of f [1 pt.] ii. Show that the function is neither surjective nor injective. [2 pt.] iii. Now choose two sets X and Y such that f restricted to X is 3 pt.] invertible. Write down f iv. Consider the function h :R-R given by h(x)-. What is the composition ho f h(f(x))? What is the composition 3 pt.] foh- f(h(x))? (c) Four letters are initially written in the following order: ITCY i. Write down in tabular form, using the numbers 1 to 4, the permu tation ?1 that rearranges these letters to forrn the word "CITY' 3 pt.] ii. What is meant by the terms transposition and fired points in the [2 pt.] context of permutations? 111. Use the cycle form of ?1 to express q as a composition of trans- [3 pt.] positions. Is ?i even or odd? iv. Consider the permutation ?,-(1234), what is ?2???? What do these composed permutation do to the original word "ITCY"? [3 pt.] v. How many total permutations are there on the 4-letter word? Within these do two permutations always commute? 2 pt.]

Answer 1

The function is f is said to be injective (one-to-one) when each element of the co-domain has a unique pre- image in the domain, i.e. if x and y are two elements in the domain and f(x) = f(y), then x = y. The function f is said to be onto, if each element of the co-domain has a at least one pre- image in the domain. The function f is said to be bijective if it is both injective and surjective. A function is invertible if it is bijective. The domain of f is R, the set of all real numbers i.e. (-infinity, infinity) as x can take all real values. The range of f is the set of non-negative real numbers, i.e. [0, infinity) as f(x) is always positive. Let f(x) = f(y). Then (1 - x)^2= (1 - y)^2 so that (1 - x) = plusminus(l - y). Then x = y or, x = 2 - y.