Answer :
To determine the end behavior of the function [tex]\( g \)[/tex] and compare it to another function [tex]\( f \)[/tex], we should closely examine the provided values in the table for [tex]\( g \)[/tex]. Let's first state the values provided in [tex]\( g(x) \)[/tex] at the given [tex]\( x \)[/tex]-coordinates:
[tex]\[ \begin{array}{c|c|c|c|c|c} x & -1 & 0 & 1 & 3 & 4 \\ \hline g(x) & 2 & 4 & 6 & -2/3 & -2\frac{8}{9} \\ \end{array} \][/tex]
Here, [tex]\( -2\frac{2}{3} \)[/tex] is the same as [tex]\(-2.67\)[/tex], and [tex]\( -2\frac{8}{9} \)[/tex] is the same as [tex]\( -2.89 \)[/tex].
By examining the values of [tex]\( g(x) \)[/tex]:
- At [tex]\( x = -1 \)[/tex], [tex]\( g(x) = 2 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( g(x) = 4 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( g(x) = 6 \)[/tex]
- At [tex]\( x = 3 \)[/tex], [tex]\( g(x) = -\frac{2}{3} \approx -0.67 \)[/tex]
- At [tex]\( x = 4 \)[/tex], [tex]\( g(x) = -\frac{26}{9} \approx -2.89 \)[/tex]
From these values, it can be noted that as [tex]\( x \)[/tex] increases, [tex]\( g(x) \)[/tex] seems to initially increase and then starts decreasing.
Now let's analyze the complete range behavior for large values of [tex]\( x \)[/tex] both positively and negatively:
1. As [tex]\( x \to \infty \)[/tex]:
- [tex]\( g(x) \)[/tex] begins to show a decreasing trend based on the negative values at [tex]\( x = 3 \)[/tex] and [tex]\( x = 4 \)[/tex]. Therefore, [tex]\( g(x) \)[/tex] likely approaches decreasing infinitely.
2. As [tex]\( x \to -\infty \)[/tex]:
- There isn't enough data explicitly within the table for large negative [tex]\( x \)[/tex], but based on the pattern provided, we may infer the behavior from trends or theoretical expectations outside the provided data points.
Considering the end behavior for the function:
- Both towards [tex]\( \infty \)[/tex] and [tex]\( -\infty \)[/tex], even if the data is sparse towards negative infinity, the overall trend of decrease can possibly mean that [tex]\( g(x) \)[/tex] follows similar decreasing behavior towards both infinities.
However, for any precise statement, comparison with another function [tex]\( f \)[/tex] based on this assumption would be apparent on the basis of approaching infinity or negative infinity uniformly in respect [tex]\( g \)[/tex].
Since concrete information from [tex]\( -\infty \)[/tex] is lacking explicitly, one could analyze overall trends:
From all observations, Option A reflects the uniformity in decreasing trend evident numerically here:
Therefore the answer is:
A. They have the same end behavior as [tex]\(x\)[/tex] approaches [tex]\( -\infty \)[/tex] and the same end behavior as [tex]\(x\)[/tex] approaches [tex]\( \infty\)[/tex].
[tex]\[ \begin{array}{c|c|c|c|c|c} x & -1 & 0 & 1 & 3 & 4 \\ \hline g(x) & 2 & 4 & 6 & -2/3 & -2\frac{8}{9} \\ \end{array} \][/tex]
Here, [tex]\( -2\frac{2}{3} \)[/tex] is the same as [tex]\(-2.67\)[/tex], and [tex]\( -2\frac{8}{9} \)[/tex] is the same as [tex]\( -2.89 \)[/tex].
By examining the values of [tex]\( g(x) \)[/tex]:
- At [tex]\( x = -1 \)[/tex], [tex]\( g(x) = 2 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( g(x) = 4 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( g(x) = 6 \)[/tex]
- At [tex]\( x = 3 \)[/tex], [tex]\( g(x) = -\frac{2}{3} \approx -0.67 \)[/tex]
- At [tex]\( x = 4 \)[/tex], [tex]\( g(x) = -\frac{26}{9} \approx -2.89 \)[/tex]
From these values, it can be noted that as [tex]\( x \)[/tex] increases, [tex]\( g(x) \)[/tex] seems to initially increase and then starts decreasing.
Now let's analyze the complete range behavior for large values of [tex]\( x \)[/tex] both positively and negatively:
1. As [tex]\( x \to \infty \)[/tex]:
- [tex]\( g(x) \)[/tex] begins to show a decreasing trend based on the negative values at [tex]\( x = 3 \)[/tex] and [tex]\( x = 4 \)[/tex]. Therefore, [tex]\( g(x) \)[/tex] likely approaches decreasing infinitely.
2. As [tex]\( x \to -\infty \)[/tex]:
- There isn't enough data explicitly within the table for large negative [tex]\( x \)[/tex], but based on the pattern provided, we may infer the behavior from trends or theoretical expectations outside the provided data points.
Considering the end behavior for the function:
- Both towards [tex]\( \infty \)[/tex] and [tex]\( -\infty \)[/tex], even if the data is sparse towards negative infinity, the overall trend of decrease can possibly mean that [tex]\( g(x) \)[/tex] follows similar decreasing behavior towards both infinities.
However, for any precise statement, comparison with another function [tex]\( f \)[/tex] based on this assumption would be apparent on the basis of approaching infinity or negative infinity uniformly in respect [tex]\( g \)[/tex].
Since concrete information from [tex]\( -\infty \)[/tex] is lacking explicitly, one could analyze overall trends:
From all observations, Option A reflects the uniformity in decreasing trend evident numerically here:
Therefore the answer is:
A. They have the same end behavior as [tex]\(x\)[/tex] approaches [tex]\( -\infty \)[/tex] and the same end behavior as [tex]\(x\)[/tex] approaches [tex]\( \infty\)[/tex].
