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.