Question

Select the function whose end behavior is described by f left parenthesis x right parenthesis right arrow infinity as x right arrow infinity and f left parenthesis x right parenthesis right arrow negative infinity as x right arrow negative infinity.
(1 point)
Responses

f left parenthesis x right parenthesis equals 7 x superscript 9 baseline minus 3 x squared minus 6
Image with alt text: f left parenthesis x right parenthesis equals 7 x superscript 9 baseline minus 3 x squared minus 6

f left parenthesis x right parenthesis equals negative one half x cubed
Image with alt text: f left parenthesis x right parenthesis equals negative one half x cubed

f left parenthesis x right parenthesis equals x superscript 6 baseline minus 3 x cubed minus 6 x squared plus x minus 1
Image with alt text: f left parenthesis x right parenthesis equals x superscript 6 baseline minus 3 x cubed minus 6 x squared plus x minus 1

f left parenthesis x right parenthesis equals negative 5 x superscript 4 baseline minus 3 halves

Answer 1

To determine which function has the end behavior described by \( f(x) \to \infty \) as \( x \to \infty \) and \( f(x) \to -\infty \) as \( x \to -\infty \), we should consider the leading term of each polynomial function. The leading term's degree and coefficient dictate the end behavior of the polynomial as \( x \) approaches positive and negative infinity.

1. \( f(x) = 7x^9 - 3x^2 - 6 \)
- Leading term: \( 7x^9 \) (degree 9, positive coefficient)
- \( f(x) \to \infty \) as \( x \to \infty \)
- \( f(x) \to -\infty \) as \( x \to -\infty \) (since an odd-degree polynomial with a positive leading coefficient goes down to negative infinity on the left side).

2. \( f(x) = -\frac{1}{2}x^3 \)
- Leading term: \( -\frac{1}{2}x^3 \) (degree 3, negative coefficient)
- \( f(x) \to -\infty \) as \( x \to \infty \)
- \( f(x) \to \infty \) as \( x \to -\infty \)

3. \( f(x) = x^6 - 3x^3 - 6x^2 + x - 1 \)
- Leading term: \( x^6 \) (degree 6, positive coefficient)
- \( f(x) \to \infty \) as \( x \to \infty \)
- \( f(x) \to \infty \) as \( x \to -\infty \)

4. \( f(x) = -5x^4 - \frac{3}{2} \)
- Leading term: \( -5x^4 \) (degree 4, negative coefficient)
- \( f(x) \to -\infty \) as \( x \to \infty \)
- \( f(x) \to -\infty \) as \( x \to -\infty \)

The function whose end behavior matches the description \( f(x) \to \infty \) as \( x \to \infty \) and \( f(x) \to -\infty \) as \( x \to -\infty \) is:

**Answer: \( f(x) = 7x^9 - 3x^2 - 6 \)**