Arcsin(6x)

The expression \(\arcsin(6x)\) refers to the inverse sine function applied to the argument \(6x\). However, it's important to note that the arcsine function is only defined for inputs in the range of \([-1, 1]\). This means that for the expression \(\arcsin(6x)\) to be valid, the argument \(6x\) must also lie within this range.

Thus, we have the following conditions that need to be satisfied:

\[ -1 \leq 6x \leq 1 \]

Dividing the entire inequality by \(6\) yields:

\[ -\frac{1}{6} \leq x \leq \frac{1}{6} \]

So, \(\arcsin(6x)\) is defined only for \(x\) values within the interval \([-1/6, 1/6]\).

Key Points:

• \(\arcsin(6x)\) is defined for \(x\) in the interval \([-1/6, 1/6]\).
• The output of \(\arcsin(y)\) is in the range \([- \frac{\pi}{2}, \frac{\pi}{2}]\) for any \(y\) in the interval \([-1, 1]\).

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