To determine the range of the function [tex]\( f(x) = -3^x - 5 \)[/tex], let's analyze its behavior carefully.
### Step-by-Step Solution
1. Understand the Function's Components:
- The function [tex]\( f(x) = -3^x - 5 \)[/tex] involves an exponential term [tex]\( 3^x \)[/tex] and a constant term (-5).
2. Analyze the Exponential Term [tex]\( 3^x \)[/tex]:
- When [tex]\( x \to \infty \)[/tex] (as [tex]\( x \)[/tex] approaches positive infinity), [tex]\( 3^x \)[/tex] grows larger and larger without bounds. Since we have a negative sign in front of [tex]\( 3^x \)[/tex], [tex]\( -3^x \)[/tex] will approach negative infinity.
- When [tex]\( x \to -\infty \)[/tex] (as [tex]\( x \)[/tex] approaches negative infinity), [tex]\( 3^x \)[/tex] approaches 0 because any positive number raised to a very large negative power results in a number close to zero. Thus, [tex]\( -3^x \)[/tex] will approach 0.
3. Combine with the Constant Term (-5):
- As [tex]\( x \to \infty \)[/tex], since [tex]\( -3^x \)[/tex] approaches negative infinity, [tex]\( f(x) = -3^x - 5 \)[/tex] will also approach negative infinity. Specifically, [tex]\( f(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], since [tex]\( -3^x \)[/tex] approaches 0, [tex]\( f(x) = -3^x - 5 \)[/tex] will approach [tex]\(-5\)[/tex]. Specifically, [tex]\( f(x) \to -5\)[/tex].
4. Determine the Range:
- We observe the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] varies over all real numbers. The function values will continue to decrease without bound as [tex]\( x \to \infty \)[/tex], and the values will approach but never equal [tex]\(-5\)[/tex] as [tex]\( x \to -\infty \)[/tex].
- Therefore, the function [tex]\( f(x) \)[/tex] takes on all real numbers less than [tex]\(-5\)[/tex].
### Conclusion
The range of the function [tex]\( f(x) = -3^x - 5 \)[/tex] is the set of real numbers less than [tex]\(-5\)[/tex]. Thus, the correct answer is:
The set of real numbers less than [tex]\(-5\)[/tex].
