y = (-3x^(2)cos(ln(x^(3)+1)))/(x^(3 )+1) find domain and range

To find the domain and range of the function:

\[
y = \frac{-3x^2 \cos(\ln(x^3 + 1))}{x^3 + 1}
\]

### 1. **Domain**:
The domain of a function consists of all \( x \)-values for which the function is defined.

- The only part of the expression that could restrict the domain is \( \ln(x^3 + 1) \) and the denominator \( x^3 + 1 \).
- For \( \ln(x^3 + 1) \) to be defined, the argument \( x^3 + 1 \) must be strictly positive. Since \( x^3 + 1 \geq 1 \) for all real values of \( x \), the logarithm is defined for all \( x \).
- The denominator \( x^3 + 1 \) cannot be zero. We solve \( x^3 + 1 = 0 \), which gives \( x = -1 \). Therefore, \( x = -1 \) must be excluded from the domain.

Thus, the domain is:
\[
\text{Domain} = (-\infty, -1) \cup (-1, \infty)
\]

### 2. **Range**:
The range consists of all possible values of \( y \). To analyze this:

- The numerator \( -3x^2 \cos(\ln(x^3 + 1)) \) includes a cosine term, which oscillates between \( -1 \) and \( 1 \).
- Since \( \cos(\ln(x^3 + 1)) \) oscillates between \( -1 \) and \( 1 \), the factor \( -3x^2 \) will affect the amplitude.
  - For large positive and negative values of \( x \), \( x^3 + 1 \) grows rapidly, causing the denominator to dominate and the function to approach zero.
  - For small values of \( x \), the cosine can still take values between \( -1 \) and \( 1 \), meaning that \( y \) will oscillate between positive and negative values for non-zero \( x \).

Given that the function is continuous (except at \( x = -1 \)) and approaches zero for large values of \( x \), the range is:

\[
\text{Range} = (-\infty, \infty)
\]

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Would you like further explanation on any step?

Here are some related questions:
1. How do you determine the domain of functions involving logarithms?
2. What are the properties of cosine functions and how do they affect the range?
3. Can a function have holes or discontinuities in its domain?
4. How does the behavior of a function as \( x \to \infty \) help in finding its range?
5. What happens when the denominator of a function includes terms like \( x^3 + 1 \)?

**Tip**: When working with logarithmic functions, always check that the argument of the logarithm is positive to determine the domain.

Explore how to determine the domain and range of the function y = (-3x^2 cos(ln(x^3 + 1)))/(x^3 + 1). Learn about logarithmic and trigonometric functions, domain restrictions, and the behavior of rational functions. This problem is suited for students in Grades 11-12 or early college calculus.

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