In the interval (1

3. Domain for (x > 2):

For (x > 2), both the numerator (sin(x)) and the denominator (sqrt{x 2} - frac{1}{x-2}) are well-defined and positive. Therefore, the function is defined in the interval ((2, infty)).

Combining these findings, the domain of the function is:

mathbb{R} - left{ -infty, -2, 1, 2 right}

Range of the Function

The range of the function is determined by the values that the function can take. Since (sin(x)) oscillates between (-1) and (1), and the denominator (sqrt{x 2} - frac{1}{x-2}) can take any real value depending on (x), the function (f(x)) will also oscillate between (-1) and (1).

Thus, the range of the function is:

code{[-1, 1]}

Complex Analysis

When we extend the function to the complex plane, things become more intriguing. In the realm of complex numbers, (z in mathbb{C}), the function's behavior changes dramatically.

1. Natural Logarithm and Complex Analysis:

The natural logarithm of a complex number, (ln(z)), is multi-valued. This means that for any complex number (z), (ln(z)) can take infinitely many values, differing by integer multiples of (2pi i).

2. Imaginary Parts and Hyperbolic Functions:

When the function involves imaginary parts or hyperbolic functions, it can lead to more complex behavior. For example, if the function includes terms like (cosh(z)) or (sinh(z)), these hyperbolic functions will have real and imaginary components.

For instance, if (f(z) sin(z) / (sqrt{z 2} - frac{1}{z-2})), where (z) is a complex number, the function will involve both real and imaginary parts, and the function's range and domain will be more complex, possibly including both real and imaginary values.

Thus, when extended to complex analysis, the function becomes:

z in mathbb{C}) implies things change dramatically! :P

In this scenario, the function's range and domain can involve both real and imaginary parts, making the analysis more intricate.

Conclusion

The domain and range of a function can provide crucial insights into its behavior and properties. In the context of the function (y f(x) sin(x) / (sqrt{x 2} - frac{1}{x-2})), we have explored how it behaves in the real number system and in the complex plane. Understanding these concepts is essential for further exploration in complex analysis and applied mathematics.

Keywords: domain, range, complex analysis, imaginary parts, hyperbolic functions