What is the end behavior of this radical function? f(x) = -2∛x+7

A. As x approaches negative infinity f(x) approaches 0.
B. As x approaches positive infinity f(x) approaches negative infinity.
C. As x approaches negative infinity f(x) approaches negative infinity.
D. As x approaches positive infinity f(x) approaches positive infinity.

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

To understand the end behavior of the function f ( x ) = − 2 3 x ​ + 7 , we'll analyze how the function behaves as x approaches positive and negative infinity.
Step-by-Step Analysis:

Understand the Function:


The function f ( x ) = − 2 3 x ​ + 7 involves a cube root term. The cube root function 3 x ​ can take on all real numbers, including negative numbers, because cube roots of negative numbers are also negative.


Behavior as x → ∞ :


As x becomes very large (approaches positive infinity), the cube root 3 x ​ also becomes large, and thus − 2 3 x ​ becomes more negative.

The term " + 7 " adds 7 to every result of − 2 3 x ​ , but as x → ∞ , the − 2 3 x ​ part dominates.

So, f ( x ) → − ∞ .



Behavior as x → − ∞ :


When x becomes very negative (approaches negative infinity), 3 x ​ becomes more negative. This is because cube roots retain the sign of the original number.

Therefore, − 2 3 x ​ becomes increasingly positive.

Again, the term " + 7 " adjusts every value by 7, but the dominating effect is still from − 2 3 x ​ .

Thus, f ( x ) → ∞ .


Conclusion:

For positive infinity: As x approaches positive infinity, f ( x ) → − ∞ .

For negative infinity: As x approaches negative infinity, f ( x ) → ∞ .


The correct choice based on this analysis is B. As x approaches positive infinity, f(x) approaches negative infinity.

Answered by EmmaGraceJohnson | 2025-07-07