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.