Let's analyze the behavior of the given function [tex]\( f(x) = -\sqrt{x + 6} \)[/tex].
1. As [tex]\( x \)[/tex] approaches positive infinity:
- When [tex]\( x \to +\infty \)[/tex], the expression [tex]\( x + 6 \)[/tex] will also approach [tex]\( +\infty \)[/tex].
- The square root of a number that approaches infinity is also approaching infinity: [tex]\( \sqrt{x + 6} \to +\infty \)[/tex].
- Since there is a negative sign in front of the square root, [tex]\( f(x) \)[/tex] will be the negative of an increasingly large positive number: [tex]\( -\sqrt{x + 6} \to -\infty \)[/tex].
Given this analysis, we can conclude:
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.
2. As [tex]\( x \)[/tex] approaches negative infinity:
- When [tex]\( x \to -\infty \)[/tex], the term [tex]\( x + 6 \)[/tex] will become a very large negative number.
- The square root of a negative number is not defined in the real numbers (it becomes a complex number).
- This renders the function [tex]\( f(x) \)[/tex] as undefined for large negative values of [tex]\( x \)[/tex].
Since the function is not defined for [tex]\( x \to -\infty \)[/tex], the respective options involving negative infinity will be incorrect.
From the given options:
A. As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
- This is incorrect because [tex]\( f(x) \to -\infty \)[/tex], not [tex]\( +\infty \)[/tex].
B. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
- This is incorrect because the function [tex]\( f(x) \)[/tex] is undefined for [tex]\( x \to -\infty \)[/tex].
C. As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.
- This is correct as per our analysis.
D. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.
- This is incorrect because the function [tex]\( f(x) \)[/tex] is undefined for [tex]\( x \to -\infty \)[/tex].
Therefore, the correct answer is:
C. As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.
