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Which function is the inverse of [tex]$f(x)=-x^3-9$[/tex]?

A. [tex]$f^{-1}(x)=\sqrt[3]{x+9}$[/tex]
B. [tex]$f^{-1}(x)=\sqrt[3]{-x-9}$[/tex]
C. [tex]$f^{-1}(x)=-\sqrt[3]{-x+9}$[/tex]
D. [tex]$f^{-1}(x)=-\sqrt[3]{x-9}$[/tex]


Sagot :

To determine the inverse function of \( f(x) = -x^3 - 9 \), we need to follow these steps:

1. Start with the original function:
[tex]\[ y = -x^3 - 9 \][/tex]

2. Swap \( x \) and \( y \) to find the inverse relation:
[tex]\[ x = -y^3 - 9 \][/tex]

3. Solve this equation for \( y \):
- First, isolate the term containing \( y \). Add 9 to both sides of the equation:
[tex]\[ x + 9 = -y^3 \][/tex]
- Next, multiply both sides by -1 to make the \( y^3 \) term positive:
[tex]\[ -(x + 9) = y^3 \][/tex]
- Finally, take the cube root of both sides to solve for \( y \):
[tex]\[ y = \sqrt[3]{-(x + 9)} \][/tex]
Which can also be written as:
[tex]\[ y = -\sqrt[3]{x + 9} \][/tex]

Therefore, the inverse function of \( f(x) = -x^3 - 9 \) is:

[tex]\[ f^{-1}(x) = -\sqrt[3]{x + 9} \][/tex]

So, the correct answer is:

[tex]\[ \boxed{A. \ f^{-1}(x)=\sqrt[3]{x+9}} \][/tex]
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