To determine the end behavior of the function [tex]\( g(x) = 4|x - 2| - 3 \)[/tex], let's analyze how the function behaves as [tex]\( x \)[/tex] approaches negative infinity and positive infinity:
1. Behavior as [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]):
- The expression [tex]\(|x - 2|\)[/tex] measures the distance between [tex]\( x \)[/tex] and 2. As [tex]\( x \)[/tex] becomes very large and negative, the distance [tex]\(|x - 2|\)[/tex] also becomes very large.
- Therefore, [tex]\(|x - 2|\)[/tex] increases without bound as [tex]\( x \)[/tex] approaches negative infinity.
- Since [tex]\(|x - 2|\)[/tex] is large, multiplying by 4 makes [tex]\( 4|x - 2| \)[/tex] grow even larger.
- Subtracting 3 doesn't significantly affect the dominant behavior of [tex]\( 4|x - 2| \)[/tex] as [tex]\( x \to -\infty \)[/tex].
- Hence, [tex]\( g(x) = 4|x - 2| - 3 \)[/tex] will increase without bound as [tex]\( x \)[/tex] approaches negative infinity.
- So, as [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \)[/tex] approaches positive infinity.
2. Behavior as [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to \infty \)[/tex]):
- Similarly, as [tex]\( x \)[/tex] becomes very large and positive, the distance [tex]\(|x - 2|\)[/tex] will again become very large.
- Thus, [tex]\(|x - 2|\)[/tex] increases without bound as [tex]\( x \)[/tex] approaches positive infinity.
- Multiplying by 4 makes [tex]\( 4|x - 2| \)[/tex] grow larger, and subtracting 3 affects the growth insignificantly.
- Therefore, [tex]\( g(x) = 4|x - 2| - 3 \)[/tex] will increase without bound as [tex]\( x \)[/tex] approaches positive infinity.
- So, as [tex]\( x \to \infty \)[/tex], [tex]\( g(x) \)[/tex] approaches positive infinity.
Conclusively, the correct selections are:
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( g(x) \)[/tex] approaches positive infinity.
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( g(x) \)[/tex] approaches positive infinity.
