what is the domain and range of cos (ln(x^3 + 1))

To find the domain and range of the function \( f(x) = \cos(\ln(x^3 + 1)) \), let's go step by step:

### Domain:
The domain of a function is the set of all possible input values (x-values) for which the function is defined.

1. The argument inside the logarithm must be positive, because \( \ln(y) \) is defined only when \( y > 0 \).
   
   So, we need:
   \[
   x^3 + 1 > 0
   \]
   \[
   x^3 > -1
   \]
   Since \( x^3 \geq -1 \) for all real \( x \), the logarithmic function is defined for all real values of \( x \). Thus, the domain of the function is:
   \[
   \text{Domain: } (-\infty, \infty)
   \]

### Range:
The range is the set of possible output values (y-values) for the function.

2. The cosine function, \( \cos(\theta) \), always takes values between \(-1\) and \(1\), for any real argument \( \theta \). This holds true regardless of the value inside the cosine function, whether it is a logarithmic expression or not.

   Therefore, since \( \cos(\theta) \) is always bounded by \(-1\) and \(1\), the range of \( f(x) = \cos(\ln(x^3 + 1)) \) is:
   \[
   \text{Range: } [-1, 1]
   \]

### Final Answer:
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( [-1, 1] \)

Would you like more details on any part of this solution?

Here are some related questions:
1. What is the domain and range of \( \ln(x^2 + 1) \)?
2. How does the behavior of \( \ln(x) \) affect composite functions?
3. What are the properties of the cosine function over all real numbers?
4. Can the composition of \( \ln(x) \) and trigonometric functions yield different ranges?
5. How do you determine the range for a function involving both trigonometric and logarithmic expressions?

**Tip:** When finding the domain of a logarithmic function, always check where the argument inside the logarithm is greater than zero.

What is the domain and range of cos(ln(x^3 + 1))?

Discover how to find the domain and range of the composite function cos(ln(x^3 + 1)). This solution covers logarithmic and trigonometric functions, their properties, and how they combine. Suitable for students in Grades 11-12.

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