To determine the correctness of the statements about the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], let's go through each statement one by one.
Given the functions:
[tex]\[ f(x) = 3x^3 + 2 \][/tex]
[tex]\[ g(x) = \sqrt[3]{\frac{x-2}{3}} \][/tex]
Statement I: The function [tex]\( f(g(x)) = x \)[/tex] for all real [tex]\( x \)[/tex].
To verify [tex]\( f(g(x)) \)[/tex]:
1. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(g(x)) = f\left( \sqrt[3]{\frac{x-2}{3}} \right) \][/tex]
2. Apply the definition of [tex]\( f \)[/tex]:
[tex]\[ f\left( \sqrt[3]{\frac{x-2}{3}} \right) = 3 \left( \sqrt[3]{\frac{x-2}{3}} \right)^3 + 2 \][/tex]
3. Simplify the expression:
[tex]\[ 3 \left( \frac{x-2}{3} \right) + 2 = x - 2 + 2 = x \][/tex]
So, [tex]\( f(g(x)) = x \)[/tex] holds for all real [tex]\( x \)[/tex].
Statement II: The function [tex]\( g(f(x)) = x \)[/tex] for all real [tex]\( x \)[/tex].
To verify [tex]\( g(f(x)) \)[/tex]:
1. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(f(x)) = g(3x^3 + 2) \][/tex]
2. Apply the definition of [tex]\( g \)[/tex]:
[tex]\[ g(3x^3 + 2) = \sqrt[3]{\frac{3x^3 + 2 - 2}{3}} \][/tex]
3. Simplify the expression:
[tex]\[ \sqrt[3]{\frac{3x^3}{3}} = \sqrt[3]{x^3} = x \][/tex]
So, [tex]\( g(f(x)) = x \)[/tex] holds for all real [tex]\( x \)[/tex].
Statement III: Functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverse functions.
From the results of Statements I and II, both [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex] hold for all real [tex]\( x \)[/tex]. Therefore, [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are indeed inverse functions of each other.
Thus, all the statements I, II, and III are true, and the correct answer is:
C. I, II, and III
