Bolton Gibson
12/16/2023 · Middle School
Consider the functions \( f(x)=x^{3}-3 \) and \( g(x)=\sqrt[3]{x+3} \). (a) Find \( f(g(x)) \). (b) Find \( g(f(x)) \). (c) Determine whether the functions \( f \) and \( g \) are inverses of each other.
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(a) To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f\left(\sqrt[3]{x+3}\right) = \left(\sqrt[3]{x+3}\right)^3 - 3 = x + 3 - 3 = x. \]
(b) To find \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(x^3 - 3) = \sqrt[3]{(x^3 - 3) + 3} = \sqrt[3]{x^3} = x. \]
(c) To determine if \( f \) and \( g \) are inverses of each other, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \) in the domains of both functions. We have already shown that both \( f(g(x)) = x \) and \( g(f(x)) = x \) are true. Therefore, \( f \) and \( g \) are inverses of each other.
Quick Answer
(a) \( f(g(x)) = x \).
(b) \( g(f(x)) = x \).
(c) \( f \) and \( g \) are inverses of each other.
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