Given that, f(x) = x2 + 2 and, g(x) = x2 - 3
(a) (fog)(x) = f(g(x))
i.e. (fog)(x) = f(x2 - 3)
i.e. (fog)(x) = (x2 - 3)2 + 2
i.e. (fog)(x) = x4 - 6x2 + 9 + 2
i.e. (fog)(x) = x4 - 6x2 + 11
.
(b) (gof)(x) = g(f(x))
i.e. (gof)(x) = g(x2 + 2)
i.e. (gof)(x) = (x2 + 2)2 - 3
i.e. (gof)(x) = x4 + 4x2 + 4 - 3
i.e. (gof)(x) = x4 + 4x2 + 1
.
(c) (fog)(3) = 34 - (6 * 32) + 11 = 81 - 54 + 11 = 38
.
(d) (gof)(3) = 34 + (4 * 32) + 1 = 118
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