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Select the correct answer.

What is the inverse of this function?

[tex]\[ f(x)=\sqrt[3]{x+12} \][/tex]

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]


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].