Let's solve each equation step by step, given $f(x) = \sqrt[3]{x}$.

1. $(f(x))^{-1} = 11$

We know that $f(x) = \sqrt[3]{x}$, so:

$(f(x))^{-1} = \frac{1}{f(x)} = 11$

This implies:

$\frac{1}{\sqrt[3]{x}} = 11$

Now, solve for $x$:

$\sqrt[3]{x} = \frac{1}{11}$

Cube both sides:

$x = \left(\frac{1}{11}\right)^3 = \frac{1}{11^3} = \frac{1}{1331}$

Thus, $x = \frac{1}{1331}$.

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2. $f^{-1}(x) = 11$

$f^{-1}(x)$ represents the inverse of $f(x)$, where $f(x) = \sqrt[3]{x}$. The inverse of $f(x)$ is $f^{-1}(x) = x^3$.

Given:

$f^{-1}(x) = 11$

This means:

$x^3 = 11$

Solve for $x$:

$x = \sqrt[3]{11}$

So, $x = \sqrt[3]{11}$ (or approximately $x \approx 2.223$).

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3. $f(x^{-1}) = 11$

We know $f(x) = \sqrt[3]{x}$, so:

[ f(x^{-1}) = \sqrt[3]{x^{-1}} = \frac{1}{\sqrt[3]{x}} ]

Given:

$\frac{1}{\sqrt[3]{x}} = 11$

This is the same as the first equation:

$\sqrt[3]{x} = \frac{1}{11}$

Cube both sides:

$x = \frac{1}{11^3} = \frac{1}{1331}$

Thus, $x = \frac{1}{1331}$.

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Summary of Solutions:

1. $(f(x))^{-1} = 11 \Rightarrow x = \frac{1}{1331}$
2. $f^{-1}(x) = 11 \Rightarrow x = \sqrt[3]{11}$
3. $f(x^{-1}) = 11 \Rightarrow x = \frac{1}{1331}$

Would you like more details or explanations?

Here are 5 related questions to expand on this:

1. How do you calculate the inverse of a function?
2. What are the properties of cube roots in solving equations?
3. How do you solve equations with exponents in the denominator?
4. What is the significance of a function’s inverse in algebra?
5. How do you interpret and solve $f(x^{-1})$?

Tip: Always check whether you need to take the inverse of the function or the inverse of the variable before solving.