Looking for reliable answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
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.
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
