Compositions of FunctionsConsider functions, f: A → B and g: B → C. The composition of f with g is a function from A into C defined by (gof) (x) = g [f(x)] and is defined by gof. To find the composition of f and g, first find the image of x under f and then find the image of f (x) under g. Example1: Consider the function f = {(1, a), (2, a), (3, b)} and g = {(a, 5), (b, 7)} as in figure. Find the composition of gof. Solution: The composition function gof is shown in fig: (gof) (1) = g [f (1)] = g (a) = 5, (gof) (2) = g [f (2)] = g (a) = 5 (gof) (3) = g [f (3)] = g (b) = 7. Example2: Consider f, g and h, all functions on the integers, by f (n) =n^{2}, g (n) = n + 1 and h (n) = n  1. Determine (i) hofog (ii) gofoh (iii) fogoh. Solution: (i) hofog (n) = n + 1, hofog (n + 1) = (n+1)^{2} h [(n+1)^{2} ] = (n+1)^{2}  1 = n^{2} + 1 + 2n  1 = n^{2} + 2n. (ii) gofoh (n) = n  1, gof (n  1) = (n1)^{2} g [(n1)^{2} ] = (n1)^{2} + 1 = n^{2} + 1  2n + 1 = n^{2}  2n + 2. (iii) fogoh (n) = n  1 fog (n  1) = (n  1) + 1 f (n) = n^{2}. Note:
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