From Chapter 3 - Functions and Algorithms; select one problem from the Solved Problemslist. Explain the problem and present the solution also explaining (in detail)each step and why the solution is correct.

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76 of 490 Q - + CL A | | V V E 60 FUNCTIONS AND ALGORITHMS [CHAP. 3 Solved Problems FUNCTIONS 3.1. Let X = [1, 2, 3, 4). Determine whether each relation on X is a function from X into X. (a) f = {(2, 3), (1, 4), (2, 1), (3.2), (4, 4)} (b) g = {(3, 1), (4, 2). (1, 1)} (c) h = {(2, 1), (3, 4), (1, 4), (2, 1), (4,4) } Recall that a subset fof X x X is a function f: X -> X if and only if each a e X appears as the first coordinate in exactly one ordered pair in f. (a) No. Two different ordered pairs (2, 3) and (2, 1) in fhave the same number 2 as their first coordinate. (b) No. The element 2 e X does not appear as the first coordinate in any ordered pair in g. (c) Yes. Although 2 6 X appears as the first coordinate in two ordered pairs in h, these two ordered pairs are equal. 3.2. Sketch the graph of: (a) f(x) = x-+x-6; (b) g(x) = x3- 3x2 -x+3. Set up a table of values for x and then find the corresponding values of the function. Since the functions are polynomials, plot the points in a coordinate diagram and then draw a smooth continuous curve through the points. See Fig. 3-8. S (x) -2 -15 - -12 Graph of f - 12 + x-6 Graph of g - x3 - 3x3 - x + 3 Fig. 3-8 3.3. Let A = (a, b, c), B = (x, y, z], C = {r, s, f). Let f: A - B and g: B - C be defined by: f = {(a, y)(b, x), (c, y)] and g = ((x, $), (y, 1), (z, r)]