We are given the function $f(x) = x^3 - \frac{10}{x}$. Let's compute each of the requested values step by step.

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(a) $f\left(-\frac{1}{2}\right)$

Substitute $x = -\frac{1}{2}$ into the function: $f\left(-\frac{1}{2}\right) = \left(-\frac{1}{2}\right)^3 - \frac{10}{-\frac{1}{2}}$

Step-by-step:

1. $\left(-\frac{1}{2}\right)^3 = -\frac{1}{8}$,
2. $-\frac{10}{-\frac{1}{2}} = 10 \cdot 2 = 20$,
3. Combine the terms: $-\frac{1}{8} + 20 = \frac{-1 + 160}{8} = \frac{159}{8}$.

Thus, $f\left(-\frac{1}{2}\right) = \frac{159}{8}$.

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(b) $f(2)$

Substitute $x = 2$ into the function: $f(2) = 2^3 - \frac{10}{2}$

Step-by-step:

1. $2^3 = 8$,
2. $-\frac{10}{2} = -5$,
3. Combine the terms: $8 - 5 = 3$.

Thus, $f(2) = 3$.

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(c) $f(-2)$

Substitute $x = -2$ into the function: $f(-2) = (-2)^3 - \frac{10}{-2}$

Step-by-step:

1. $(-2)^3 = -8$,
2. $-\frac{10}{-2} = 5$,
3. Combine the terms: $-8 + 5 = -3$.

Thus, $f(-2) = -3$.

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Final Answers:

(a) $f\left(-\frac{1}{2}\right) = \frac{159}{8}$
(b) $f(2) = 3$
(c) $f(-2) = -3$

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Would you like further details or clarification?
Here are 5 related questions to explore:

1. How is the domain of $f(x)$ affected by the term $\frac{10}{x}$?
2. What is the derivative $f'(x)$ of this function?
3. How would the function behave as $x \to \infty$ and $x \to 0$?
4. What are the roots of $f(x)$, if any?
5. Can you determine the critical points of $f(x)$?

Tip: Always check for undefined points in rational functions where the denominator is zero!