Describe the end behavior of the function f left parenthesis x right parenthesis equals negative 2 x superscript 4 baseline minus x cubed plus 3.

Question

(1 point)
Responses

f left parenthesis x right parenthesis right arrow infinity as x right arrow negative infinity; f left parenthesis x right parenthesis right arrow infinity as x right arrow infinity
Image with alt text: f left parenthesis x right parenthesis right arrow infinity as Image with alt text: x right arrow negative infinity ; Image with alt text: f left parenthesis x right parenthesis right arrow infinity as Image with alt text: x right arrow infinity

f left parenthesis x right parenthesis right arrow negative infinity as x right arrow negative infinity; f left parenthesis x right parenthesis right arrow negative infinity as x right arrow infinity
Image with alt text: f left parenthesis x right parenthesis right arrow negative infinity as Image with alt text: x right arrow negative infinity ; Image with alt text: f left parenthesis x right parenthesis right arrow negative infinity as Image with alt text: x right arrow infinity

f left parenthesis x right parenthesis right arrow infinity as x right arrow negative infinity; f left parenthesis x right parenthesis right arrow negative infinity as x right arrow infinity
Image with alt text: f left parenthesis x right parenthesis right arrow infinity as Image with alt text: x right arrow negative infinity ; Image with alt text: f left parenthesis x right parenthesis right arrow negative infinity as Image with alt text: x right arrow infinity

f left parenthesis x right parenthesis right arrow negative infinity as x right arrow negative infinity; f left parenthesis x right parenthesis right arrow infinity as x right arrow infinity

Answer 1

To analyze the end behavior of the function \( f(x) = -2x^4 - x^3 + 3 \), we will focus on the leading term since it will dominate the function's behavior as \( x \) approaches positive and negative infinity.

1. Leading term: The leading term in \( f(x) \) is \( -2x^4 \).

2. As \( x \to -\infty \):

   • The term \( x^4 \) becomes very large (positive), but since it is multiplied by \(-2\), \( -2x^4 \to -\infty\).
   • Thus, \( f(x) \to -\infty \) as \( x \to -\infty \).

3. As \( x \to \infty \):

   • Again, \( x^4 \) becomes very large (positive), and thus \( -2x^4 \to -\infty\).
   • Therefore, \( f(x) \to -\infty \) as \( x \to \infty \).

Based on this analysis, the end behavior of the function is:

• \( f(x) \to -\infty \) as \( x \to -\infty \)
• \( f(x) \to -\infty \) as \( x \to \infty \)

The correct response is: f(x) → negative infinity as x → negative infinity; f(x) → negative infinity as x → infinity.