To determine the end behavior of the radical function [tex]\( f(x) = 4 \sqrt{x - 6} \)[/tex], we need to analyze how [tex]\( f(x) \)[/tex] behaves as [tex]\( x \)[/tex] approaches positive infinity and negative infinity.
1. Consider [tex]\( x \)[/tex] approaching positive infinity:
- The expression inside the square root [tex]\( \sqrt{x - 6} \)[/tex].
- As [tex]\( x \)[/tex] becomes very large (i.e., [tex]\( x \)[/tex] approaches positive infinity), [tex]\( x - 6 \)[/tex] also becomes very large.
- The square root of a very large number [tex]\(\sqrt{x - 6}\)[/tex] will also become very large.
- Multiplying this large value by 4 will yield an even larger number.
Thus, as [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) = 4 \sqrt{x - 6} \)[/tex] also approaches positive infinity.
2. Consider [tex]\( x \)[/tex] approaching negative infinity:
- If [tex]\( x \)[/tex] is a very large negative number, the expression [tex]\( \sqrt{x - 6} \)[/tex] is not defined because the value inside the square root becomes negative. The square root function requires non-negative inputs.
Thus, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] is not defined.
Based on this analysis, the correct answer is:
A. As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
