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Understanding the domain and range of a function is crucial when dealing with inverse trigonometric functions.
The domain refers to all possible input values (x-values) for which the function is defined.
The range refers to all possible output values (y-values) of the function.
In simpler terms, the domain is what you can put into the function, and the range is what you get out of it.

For the function \( f(x) = \text{sin}^{-1}(3x) \), the domain is given as \( -\frac{1}{3} \leq x \leq \frac{1}{3} \). This is because \( \text{sin}^{-1} \) (also known as \( \text{arcsin} \)) has a limited range of input values where it is defined.
The range of \( f(x) \) is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), since the \( \text{arcsin} \) function only produces output values within this interval.

For the inverse function \( f^{-1}(x) \), derived to be \( \frac{\sin(x)}{3} \), the domain and range swap places:
- The domain of \( f^{-1}(x) \) becomes the range of \( f(x) \), which is \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).
- The range of \( f^{-1}(x) \) is from \( -\frac{1}{3} \) to \( \frac{1}{3} \), corresponding to the domain of \( f(x) \).

The sine function is a fundamental trigonometric function, often denoted as \( \sin(x) \). It is defined on the entire set of real numbers, but its values fall within a specific range.
For any input \( x \), \( \sin(x) \) will produce an output between -1 and 1 inclusive.

The primary properties of the sine function include:
- **Periodicity**: The function repeats its values every \( 2\pi \) interval. This is called its period.
- **Symmetry**: The function is odd, meaning \( \sin(-x) = -\sin(x) \).
- **Amplitude**: The height of the wave, or its maximum value, is 1, and the minimum is -1.

In context with our initial problem, the sine function is essential to find the inverse function \( f^{-1}(x) = \frac{\sin(x)}{3} \).
For the relevant domain of \( f(x) \), \( \ text{arcsin}(3x) \), we used properties of the sine function to solve for \( y \) in terms of \( x \).

These transformations lie at the foundation of trigonometric identities and solving inverse trigonometric equations.

Inverse functions essentially 'undo' what the original function does.
For example, if a function \( f \) maps x to y, then its inverse \( f^{-1} \) maps y back to x.
With trigonometric functions, such as sine, its inverse \( \text{arcsin} \) or \( \text{sin}^{-1} \) returns the angle whose sine value is the given number.

To find the inverse function, the steps are usually the same:
1. Start with the original function.\( f(x) = \text{sin}^{-1}(3x) \).
2. Express it in terms of \( y \): \( y = \text{sin}^{-1}(3x) \).
3. Swap \( x \) and \( y \), to solve for \( x \) in terms of \( y \).\( x = \text{sin}^{-1}(3y) \).
4. Apply the appropriate inverse operation. Apply the sine function to both sides to get \( \text{sin}(x) = 3y \).
5. Isolate \( y \). Solve for y by dividing both sides by 3: \( y = \frac{\sin(x)}{3} \).

This process of inverting functions is significant across mathematics, providing a parallel for mapping inputs to outputs and vice versa.
When dealing with trigonometric functions, understanding inverse functions deepens our comprehension of angles and their corresponding sine values.