Identify the end behavior of the function [tex]f(x)=3 x^4+x^3-7 x^2+12[/tex]
A. As [tex]x \rightarrow-\infty, y \rightarrow-\infty[/tex], and as [tex]x \rightarrow+\infty, y \rightarrow+\infty[/tex]
B. As [tex]x \rightarrow-\infty, y \rightarrow+\infty[/tex], and as [tex]x \rightarrow+\infty, y \rightarrow+\infty[/tex]
C. As [tex]x \rightarrow-\infty, y \rightarrow-\infty[/tex], and as [tex]x \rightarrow+\infty, y \rightarrow-\infty[/tex]
D. As [tex]x \rightarrow-\infty, y \rightarrow+\infty[/tex], and as [tex]x \rightarrow+\infty, y \rightarrow-\infty[/tex]
Answer (1)
Identify the dominant term: The term with the highest degree, 3 x 4 , determines the end behavior.
As x approaches + ∞ : x 4 approaches + ∞ , so f ( x ) approaches + ∞ .
As x approaches − ∞ : x 4 approaches + ∞ (because the exponent is even), so f ( x ) approaches + ∞ .
Conclusion: As x → − ∞ , y → + ∞ , and as x → + ∞ , y → + ∞ .
Explanation
Understanding the Problem We are asked to identify the end behavior of the function f ( x ) = 3 x 4 + x 3 − 7 x 2 + 12 . The end behavior of a function describes what happens to the function's values, f ( x ) (or y ), as the input x approaches positive infinity ( x \tothe f u n c t i o n ′ s v a l u es , f(x) ( or y ) , a s t h e in p u t x a pp ro a c h es p os i t i v e in f ini t y ( x \rightarrow \infty ) an d n e g a t i v e in f ini t y ( x \rightarrow -\infty$).
Focusing on the Dominant Term To determine the end behavior of the function f ( x ) = 3 x 4 + x 3 − 7 x 2 + 12 , we focus on the term with the highest degree, which is 3 x 4 . The other terms ( x 3 , − 7 x 2 , and 12 ) will become insignificant compared to 3 x 4 as x becomes very large (in either the positive or negative direction).
End Behavior as x Approaches Positive Infinity As x approaches positive infinity ( x → ∞ ), x 4 also approaches positive infinity ( x 4 → ∞ ). Since the coefficient of x 4 is positive (3), 3 x 4 also approaches positive infinity ( 3 x 4 → ∞ ). Therefore, as x → ∞ , f ( x ) → ∞ .
End Behavior as x Approaches Negative Infinity As x approaches negative infinity ( x → − ∞ ), x 4 still approaches positive infinity ( x 4 → ∞ ) because a negative number raised to an even power is positive. Again, since the coefficient of x 4 is positive (3), 3 x 4 also approaches positive infinity ( 3 x 4 → ∞ ). Therefore, as x → − ∞ , f ( x ) → ∞ .
Final Answer In summary, as x → − ∞ , y → ∞ , and as x → ∞ , y → ∞ . This corresponds to the second option provided.
Examples
Understanding the end behavior of functions is crucial in various fields. For example, in physics, when modeling the trajectory of a projectile, knowing the end behavior helps predict its long-term path. Similarly, in economics, analyzing the end behavior of a cost function can help determine how costs will behave as production levels increase significantly. This concept provides a foundation for making informed decisions and predictions in real-world scenarios.
