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Consider the end behavior of the function [tex]g(x)=4|x-2|-3[/tex].
As [tex]x[/tex] approaches negative infinity, [tex]f(x)[/tex] approaches [blank] infinity.
As [tex]x[/tex] approaches positive infinity, [tex]f(x)[/tex] approaches [blank] infinity.
Answer (1)
As x approaches negative infinity, g ( x ) approaches positive infinity.
As x approaches positive infinity, g ( x ) approaches positive infinity.
The function g ( x ) can be rewritten as 4 x − 11 when x approaches positive infinity and − 4 x + 5 when x approaches negative infinity.
Therefore, the final answer is that as x approaches both negative and positive infinity, g ( x ) approaches in f ini t y .
Explanation
Understanding the Problem We are asked to describe the end behavior of the function g ( x ) = 4∣ x − 2∣ − 3 . This means we need to determine what happens to the value of g ( x ) as x becomes very large (approaches positive infinity) and as x becomes very small (approaches negative infinity).
Analyzing Positive Infinity First, let's consider what happens as x approaches positive infinity ( x → ∞ ). When x is a very large positive number, x − 2 will also be a large positive number. Therefore, ∣ x − 2∣ = x − 2 . So, we can rewrite the function as: g ( x ) = 4 ( x − 2 ) − 3 = 4 x − 8 − 3 = 4 x − 11 As x gets larger and larger, 4 x − 11 also gets larger and larger, approaching positive infinity.
Analyzing Negative Infinity Now, let's consider what happens as x approaches negative infinity ( x → − ∞ ). When x is a very large negative number, x − 2 will also be a large negative number. Therefore, ∣ x − 2∣ = − ( x − 2 ) = − x + 2 . So, we can rewrite the function as: g ( x ) = 4 ( − x + 2 ) − 3 = − 4 x + 8 − 3 = − 4 x + 5 As x becomes a very large negative number, − 4 x becomes a very large positive number, and thus − 4 x + 5 also approaches positive infinity.
Conclusion Therefore, as x approaches negative infinity, g ( x ) approaches positive infinity, and as x approaches positive infinity, g ( x ) approaches positive infinity.
Examples
Understanding the end behavior of functions is crucial in many real-world applications. For instance, in economics, it can help predict long-term trends in market growth or decline. In physics, it can describe the behavior of systems under extreme conditions. By analyzing the function's behavior as x approaches infinity, we can make informed decisions and predictions about the system's future state. This skill is fundamental in modeling and understanding complex phenomena.
