Select the correct answer. What is the end behavior of this radical function? [tex]$f(x)=-2 \sqrt[3]{x+7}$[/tex] A. As [tex]$x$[/tex] approaches negative infinity, [tex]$f(x)$[/tex] approaches negative infinity. B. As [tex]$x$[/tex] approaches positive infinity, [tex]$f(x)$[/tex] approaches negative infinity. C. As [tex]$x$[/tex] approaches negative infinity, [tex]$f(x)$[/tex] approaches 0. D. As [tex]$x$[/tex] approaches positive infinity, [tex]$f(x)$[/tex] approaches positive infinity.

As x approaches positive infinity, x + 7 approaches positive infinity, so 3 x + 7 ​ approaches positive infinity. Thus, f ( x ) = − 2 3 x + 7 ​ approaches negative infinity.
As x approaches negative infinity, x + 7 approaches negative infinity, so 3 x + 7 ​ approaches negative infinity. Thus, f ( x ) = − 2 3 x + 7 ​ approaches positive infinity.
Therefore, as x → ∞ , f ( x ) → − ∞ .
The correct answer is B: As x approaches positive infinity, f ( x ) approaches negative infinity.

Explanation

Understanding the Problem We are given the function f ( x ) = − 2 3 x + 7 ​ and we want to determine its end behavior. This means we need to find out what happens to f ( x ) as x approaches positive infinity and negative infinity.

Behavior as x approaches positive infinity Let's analyze the behavior of the function as x approaches positive infinity ( x → ∞ ). As x becomes very large, x + 7 also becomes very large. Therefore, 3 x + 7 ​ also becomes very large and positive. Since we are multiplying by − 2 , the function f ( x ) = − 2 3 x + 7 ​ approaches negative infinity. So, as x → ∞ , f ( x ) → − ∞ .

Behavior as x approaches negative infinity Now, let's analyze the behavior of the function as x approaches negative infinity ( x → − ∞ ). As x becomes very large in the negative direction, x + 7 also becomes very large in the negative direction. Therefore, 3 x + 7 ​ also becomes very large and negative. Since we are multiplying by − 2 , the function f ( x ) = − 2 3 x + 7 ​ approaches positive infinity. So, as x → − ∞ , f ( x ) → ∞ .

Conclusion Based on our analysis:



As x approaches positive infinity, f ( x ) approaches negative infinity.
As x approaches negative infinity, f ( x ) approaches positive infinity. Therefore, the correct answer is B.

Examples
Understanding the end behavior of functions is crucial in many real-world applications. For example, in physics, it can help predict the long-term behavior of a system, such as the decay of a radioactive substance or the motion of a damped oscillator. In economics, it can be used to model the growth of a company or the spread of a disease. By analyzing the end behavior, we can gain insights into the ultimate fate of the system being modeled. For instance, if we model the population growth with a function, understanding the end behavior helps us predict if the population will stabilize, grow indefinitely, or decline to extinction. This type of analysis is not only theoretical but also has practical implications in decision-making and planning.

Answered by GinnyAnswer | 2025-07-03

The function f ( x ) = − 2 3 x + 7 ​ approaches negative infinity as x approaches positive infinity, and approaches positive infinity as x approaches negative infinity. Therefore, the correct answer is B: As x approaches positive infinity, f ( x ) approaches negative infinity.
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Answered by Anonymous | 2025-07-04