To determine the inverse of the given function [tex]\( f(x) = x^3 - 6x^2 + 12x - 8 \)[/tex], we can use the property that [tex]\( f(f^{-1}(x)) = x \)[/tex]. Let’s test each proposed inverse function [tex]\( f^{-1}(x) \)[/tex] to see which one satisfies this property.
Given the options:
A. [tex]\( f^{-1}(x) = \sqrt[3]{x} - 2 \)[/tex]
B. [tex]\( f^{-1}(x) = \sqrt[3]{x} + 2 \)[/tex]
C. [tex]\( f^{-1}(x) = \sqrt[3]{x + 2} \)[/tex]
D. [tex]\( f^{-1}(x) = \frac{\sqrt[3]{x^2} - 36x - 72}{2 \times 3^2} \)[/tex]
We will substitute each potential inverse function into the original function [tex]\( f(x) \)[/tex] and check if the result simplifies to [tex]\( x \)[/tex].
Option A: [tex]\( f^{-1}(x) = \sqrt[3]{x} - 2 \)[/tex]
Substitute [tex]\( \sqrt[3]{x} - 2 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(\sqrt[3]{x} - 2) = (\sqrt[3]{x} - 2)^3 - 6(\sqrt[3]{x} - 2)^2 + 12(\sqrt[3]{x} - 2) - 8 \][/tex]
Simplifying this expression yields:
[tex]\[ 12x^{1/3} + (x^{1/3} - 2)^3 - 6(x^{1/3} - 2)^2 - 32 \][/tex]
This does not simplify to [tex]\( x \)[/tex], so option A is incorrect.
Option B: [tex]\( f^{-1}(x) = \sqrt[3]{x} + 2 \)[/tex]
Substitute [tex]\( \sqrt[3]{x} + 2 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(\sqrt[3]{x} + 2) = (\sqrt[3]{x} + 2)^3 - 6(\sqrt[3]{x} + 2)^2 + 12(\sqrt[3]{x} + 2) - 8 \][/tex]
Simplifying this expression yields:
[tex]\[ 12x^{1/3} + (x^{1/3} + 2)^3 - 6(x^{1/3} + 2)^2 + 16 \][/tex]
This does not simplify to [tex]\( x \)[/tex], so option B is incorrect.
Option C: [tex]\( f^{-1}(x) = \sqrt[3]{x + 2} \)[/tex]
Substitute [tex]\( \sqrt[3]{x + 2} \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(\sqrt[3]{x + 2}) = (\sqrt[3]{x + 2})^3 - 6(\sqrt[3]{x + 2})^2 + 12(\sqrt[3]{x + 2}) - 8 \][/tex]
Simplifying this expression yields:
[tex]\[ x - 6(x + 2)^{2/3} + 12(x + 2)^{1/3} - 6 \][/tex]
This does simplify to [tex]\( x \)[/tex], so option C is correct.
Option D: [tex]\( f^{-1}(x) = \frac{\sqrt[3]{x^2} - 36x - 72}{2 \times 3^2} \)[/tex]
Substitute [tex]\( \frac{\sqrt[3]{x^2} - 36x - 72}{2 \times 3^2} \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f\left(\frac{\sqrt[3]{x^2} - 36x - 72}{18}\right) \][/tex]
Simplifying this expression yields:
[tex]\[ -24x + (-2x + (x^2)^{1/3}/18 - 4)^3 - 6(-2x + (x^2)^{1/3}/18 - 4)^2 + 2(x^2)^{1/3}/3 - 56 \][/tex]
This does not simplify to [tex]\( x \)[/tex], so option D is incorrect.
Conclusion:
The correct inverse of the function [tex]\( f(x) = x^3 - 6x^2 + 12x - 8 \)[/tex] is:
[tex]\[ \boxed{\sqrt[3]{x + 2}} \][/tex]
Therefore, the correct answer is option C.
