To find the value of [tex]\( b \)[/tex] given the functions [tex]\( f(x) = x^3 - 3 \)[/tex] and [tex]\( f^{-1}(x) = \sqrt[3]{x - 3b} \)[/tex], we need to ensure that the composition of the function [tex]\( f \)[/tex] with its inverse [tex]\( f^{-1} \)[/tex] returns [tex]\( x \)[/tex]. Specifically, we want:
[tex]\[ f(f^{-1}(x)) = x \][/tex]
Start by substituting [tex]\( f^{-1}(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(f^{-1}(x)) = f(\sqrt[3]{x - 3b}) \][/tex]
We know that:
[tex]\[ f(x) = x^3 - 3 \][/tex]
Therefore, substituting [tex]\( \sqrt[3]{x - 3b} \)[/tex] for [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex]:
[tex]\[ f(\sqrt[3]{x - 3b}) = (\sqrt[3]{x - 3b})^3 - 3 \][/tex]
Simplify the expression inside the function:
[tex]\[ f(\sqrt[3]{x - 3b}) = x - 3b - 3 \][/tex]
To satisfy [tex]\( f(f^{-1}(x)) = x \)[/tex], the right-hand side must equal [tex]\( x \)[/tex]:
[tex]\[ x - 3b - 3 = x \][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ -3b - 3 = 0 \][/tex]
Solve for [tex]\( b \)[/tex]:
[tex]\[ -3b = 3 \][/tex]
[tex]\[ b = -1 \][/tex]
Therefore, the correct value of [tex]\( b \)[/tex] is:
[tex]\[ \boxed{-1} \][/tex]
