If \(h(x) = x^{3} - \frac{1}{x^{3}}\), evaluate \(h(a) - h(\frac{1}{a})\).
Question 1 Report
If \(h(x) = x^{3} - \frac{1}{x^{3}}\), evaluate \(h(a) - h(\frac{1}{a})\).
Answer Details
We can start by computing the values of \(h(a)\) and \(h(\frac{1}{a})\), and then find their difference.
First, we have \(h(a) = a^{3} - \frac{1}{a^{3}}\).
Next, we have \(h(\frac{1}{a}) = (\frac{1}{a})^{3} - \frac{1}{(\frac{1}{a})^{3}} = \frac{1}{a^{3}} - a^{3}\).
Therefore,
\begin{align*}
h(a) - h(\frac{1}{a}) &= \left(a^{3} - \frac{1}{a^{3}}\right) - \left(\frac{1}{a^{3}} - a^{3}\right) \\
&= a^{3} - \frac{1}{a^{3}} - \frac{1}{a^{3}} + a^{3} \\
&= 2a^{3} - \frac{2}{a^{3}}.
\end{align*}
So the answer is \(2a^{3} - \frac{2}{a^{3}}\). Therefore, is the correct answer.
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