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What is the inverse of this function?

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

A. [tex] f^{-1}(x)=x+12 [/tex]

B. [tex] f^{-1}(x)=x-12 [/tex]

C. [tex] f^{-1}(x)=12-x^3 [/tex]

D. [tex] f^{-1}(x)=x^3-12 [/tex]

Sagot :

GinnyAnswer
To find the inverse of the function \( f(x) = \sqrt[3]{x + 12} \), we need to follow these steps:

1. Rewrite the function as an equation with \( y \) and \( x \):
Let \( y = f(x) \). So, we have:
[tex]\[ y = \sqrt[3]{x + 12} \][/tex]

2. Switch the variables \( x \) and \( y \), reflecting the fact that we want the input and output exchanged:
[tex]\[ x = \sqrt[3]{y + 12} \][/tex]

3. Eliminate the cube root by cubing both sides of the equation:
[tex]\[ x^3 = y + 12 \][/tex]

4. Solve for \( y \) to isolate it:
Subtract 12 from both sides:
[tex]\[ y = x^3 - 12 \][/tex]

So, the inverse function is:
[tex]\[ f^{-1}(x) = x^3 - 12 \][/tex]

Thus, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
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