Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Ask your questions and receive precise answers from experienced professionals across different disciplines. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
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.
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.
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
