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To find the range of the function \( f(x) = -3^x - 1 \), let’s analyze its behavior step-by-step.

1. **Base Function**: The expression \( 3^x \) is an exponential function, which has a range of all positive real numbers \( (0, \infty) \).
2. **Negative Transformation**: By adding the negative sign, we get \( -3^x \). This transformation flips the range of \( 3^x \) around the x-axis, making the range \( (-\infty, 0) \).
3. **Vertical Shift**: Subtracting 1 shifts the entire range of \( -3^x \) downward by 1, resulting in a new range of \( (-\infty, -1) \).

### Conclusion
The range of \( f(x) = -3^x - 1 \) is the set of real numbers less than \(-1\). Thus, the correct answer is:
- **The set of real numbers less than \(-1\)**.

Would you like further details or explanations?

Here are some related questions to delve deeper:
1. How does the range of an exponential function change with vertical shifts?
2. What is the range of \( f(x) = 3^x + 2 \)?
3. How does a negative exponent affect the range of exponential functions?
4. What transformations are applied to obtain \( f(x) = -3^x - 1 \) from \( 3^x \)?
5. How does the range of \( 3^x \) differ from \( -3^x \)?

**Tip**: Vertical shifts and reflections are key transformations to understand range changes in functions.

Learn how to find the range of the exponential function f(x) = -3^x - 1. This problem explores the transformations of exponential functions, including reflections and vertical shifts, to determine the function's range. Suitable for high school students.

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