Find the functions and their domains. (Enter the domains in interval notation.)f(x) = x + 1x, g(x) = x + 5x + 2(a) f ∘ g(f ∘ g)(x) = 2x2+14x+29(x+2)(x+5) domain (−∞,−5)∪(−5,−2)∪(−2,∞) (b) g ∘ f(g ∘ f)(x) = domain (c) f ∘ f(f ∘ f)(x) = domain (d) g ∘ g(g ∘ g)(x) = domain
Question
Find the functions and their domains. (Enter the domains in interval notation.)f(x) = x + 1x, g(x) = x + 5x + 2(a) f ∘ g(f ∘ g)(x) = 2x2+14x+29(x+2)(x+5) domain (−∞,−5)∪(−5,−2)∪(−2,∞) (b) g ∘ f(g ∘ f)(x) = domain (c) f ∘ f(f ∘ f)(x) = domain (d) g ∘ g(g ∘ g)(x) = domain
Solution
The question seems to be asking for the composition of functions and their domains. However, the functions for g ∘ f, f ∘ f, and g ∘ g are not provided. Here's how you can find the composition of functions and their domains:
(a) f ∘ g(x) = f(g(x)) = f(x + 5x + 2) = (x + 5x + 2) + 1/(x + 5x + 2) = 2x^2 + 14x + 29/(x + 2)(x + 5). The domain of this function is all real numbers except for -5 and -2, so in interval notation, the domain is (-∞, -5) ∪ (-5, -2) ∪ (-2, ∞).
(b) g ∘ f(x) = g(f(x)) = g(x + 1/x) = (x + 1/x) + 5(x + 1/x) + 2. To find the domain, we need to find the values of x for which the function is defined. This function is undefined when x = 0, so the domain in interval notation is (-∞, 0) ∪ (0, ∞).
(c) f ∘ f(x) = f(f(x)) = f(x + 1/x) = (x + 1/x) + 1/(x + 1/x). This function is undefined when x = 0 and x = -1, so the domain in interval notation is (-∞, -1) ∪ (-1, 0) ∪ (0, ∞).
(d) g ∘ g(x) = g(g(x)) = g(x + 5x + 2) = (x + 5x + 2) + 5(x + 5x
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