IDNLearn.com: Where questions are met with accurate and insightful answers. Join our community to access reliable and comprehensive responses to your questions from experienced professionals.
Sagot :
To find the inverse of the function [tex]\( f(x) = \sqrt[3]{x + 12} \)[/tex], we need to follow these steps:
1. Define the function: [tex]\( y = \sqrt[3]{x + 12} \)[/tex].
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]: Inverse functions are essentially a reflection over the line [tex]\( y = x \)[/tex]. So, start by swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \sqrt[3]{y + 12}. \][/tex]
3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]: Solve the equation from step 2 for [tex]\( y \)[/tex]:
[tex]\[ x = \sqrt[3]{y + 12}. \][/tex]
To isolate [tex]\( y \)[/tex], first cube both sides to get rid of the cube root:
[tex]\[ x^3 = y + 12. \][/tex]
Then, solve for [tex]\( y \)[/tex]:
[tex]\[ y = x^3 - 12. \][/tex]
4. Write the inverse function: The inverse function, denoted as [tex]\( f^{-1}(x) \)[/tex], is:
[tex]\[ f^{-1}(x) = x^3 - 12. \][/tex]
Next, we compare this result to the given choices:
- A. [tex]\( f^{-1}(x) = 12 - x^3 \)[/tex]
- B. [tex]\( f^{-1}(x) = x - 12 \)[/tex]
- C. [tex]\( f^{-1}(x) = x^3 - 12 \)[/tex]
- D. [tex]\( f^{-1}(x) = x + 12 \)[/tex]
The correct choice is clearly C: [tex]\( f^{-1}(x) = x^3 - 12 \)[/tex].
1. Define the function: [tex]\( y = \sqrt[3]{x + 12} \)[/tex].
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]: Inverse functions are essentially a reflection over the line [tex]\( y = x \)[/tex]. So, start by swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \sqrt[3]{y + 12}. \][/tex]
3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]: Solve the equation from step 2 for [tex]\( y \)[/tex]:
[tex]\[ x = \sqrt[3]{y + 12}. \][/tex]
To isolate [tex]\( y \)[/tex], first cube both sides to get rid of the cube root:
[tex]\[ x^3 = y + 12. \][/tex]
Then, solve for [tex]\( y \)[/tex]:
[tex]\[ y = x^3 - 12. \][/tex]
4. Write the inverse function: The inverse function, denoted as [tex]\( f^{-1}(x) \)[/tex], is:
[tex]\[ f^{-1}(x) = x^3 - 12. \][/tex]
Next, we compare this result to the given choices:
- A. [tex]\( f^{-1}(x) = 12 - x^3 \)[/tex]
- B. [tex]\( f^{-1}(x) = x - 12 \)[/tex]
- C. [tex]\( f^{-1}(x) = x^3 - 12 \)[/tex]
- D. [tex]\( f^{-1}(x) = x + 12 \)[/tex]
The correct choice is clearly C: [tex]\( f^{-1}(x) = x^3 - 12 \)[/tex].
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for choosing IDNLearn.com for your queries. We’re here to provide accurate answers, so visit us again soon.