Let f(x) = cos(x) and g(x)=1/x (a) Find fog and state the domain and range. (b) Find gof and state the domain and range

(a) To find fog(x), we substitute g(x) into f(x): fog(x) = f(g(x)) = f(1/x) = cos(1/x). The domain of fog(x) is all real numbers except x = 0 since g(x) has a denominator of x which cannot be zero. Therefore, the domain of fog(x) is (-∞, 0) U (0, ∞). The range of fog(x) is [-1, 1] since the cosine function ranges from -1 to 1 for all real values of x. (b) To find gof(x), we substitute f(x) into g(x): gof(x) = g(f(x)) = g(cos(x)) = 1/cos(x). The domain of gof(x) is all real numbers except for x values where cos(x) is equal to zero. Therefore, the domain of gof(x) is all real numbers except x = (2n + 1)π/2, where n is an integer. The range of gof(x) is (-∞, -1) U (1, ∞) since the reciprocal function 1/x can take on any value except zero, and cos(x) ranges from -1 to 1 for all real values of x. In summary: (a) fog(x) = cos(1/x), domain = (-∞, 0) U (0, ∞), range = [-1, 1]. (b) gof(x) = 1/cos(x), domain = all real numbers except x = (2n + 1)π/2, range = (-∞, -1) U (1, ∞).