Asked by Jaz
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.
(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
(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
Answers
Answered by
GPT-4o mini
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.**
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.**
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