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Sagot :
Let's analyze the function [tex]\( g \)[/tex] using the provided table:
[tex]\[ \begin{array}{cccccc} x & -1 & 0 & 1 & 2 & 3 \\ g(x) & -24 & -4 & 0 & \frac{4}{5} & \frac{24}{25} \end{array} \][/tex]
Step-by-Step Analysis:
1. Determining the x-intercept:
- The x-intercept is the value of [tex]\( x \)[/tex] where [tex]\( g(x) = 0 \)[/tex].
- From the table, [tex]\( g(1) = 0 \)[/tex].
- Therefore, the x-intercept is [tex]\( x = 1 \)[/tex].
2. Determining the y-intercept:
- The y-intercept is the value of [tex]\( g(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
- From the table, [tex]\( g(0) = -4 \)[/tex].
- Therefore, the y-intercept is [tex]\( y = -4 \)[/tex].
3. Analyzing the end behavior as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex]:
- To infer the end behavior, we observe the values of [tex]\( g(x) \)[/tex] as [tex]\( x \)[/tex] increases.
- As [tex]\( x \)[/tex] increases from 1 to 3, [tex]\( g(x) \)[/tex] values approach 1 (since [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{24}{25}\)[/tex] are close to 1).
- Therefore, as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], it appears that [tex]\( g(x) \)[/tex] approaches 1.
Given this analysis, let's compare the statements:
(A) "They have the same x-intercept and the same end behavior as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex]."
- Incorrect because another function must have the same x-intercept (1) and also approach 1 at [tex]\( x \)[/tex] increasing indefinitely.
(B) "They have the same end behavior as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], but they have different x- and y-intercepts."
- This statement can be true if we compare function [tex]\( g(x) \)[/tex] with another function that has the same end behavior (approaching 1), but with different x-intercepts and y-intercepts.
(C) "They have the same x-intercept and the same y-intercept."
- Incorrect because that would imply the functions have identical intercepts, which is not given in the problem.
(D) "They have the same y-intercept and the same end behavior as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex]."
- Incorrect because it is specified that the other function must have the same y-intercept as -4 and also approaches 1 as [tex]\( x \)[/tex] increases indefinitely.
The correct statement is:
(B) They have the same end behavior as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], but they have different x- and y-intercepts.
[tex]\[ \begin{array}{cccccc} x & -1 & 0 & 1 & 2 & 3 \\ g(x) & -24 & -4 & 0 & \frac{4}{5} & \frac{24}{25} \end{array} \][/tex]
Step-by-Step Analysis:
1. Determining the x-intercept:
- The x-intercept is the value of [tex]\( x \)[/tex] where [tex]\( g(x) = 0 \)[/tex].
- From the table, [tex]\( g(1) = 0 \)[/tex].
- Therefore, the x-intercept is [tex]\( x = 1 \)[/tex].
2. Determining the y-intercept:
- The y-intercept is the value of [tex]\( g(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
- From the table, [tex]\( g(0) = -4 \)[/tex].
- Therefore, the y-intercept is [tex]\( y = -4 \)[/tex].
3. Analyzing the end behavior as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex]:
- To infer the end behavior, we observe the values of [tex]\( g(x) \)[/tex] as [tex]\( x \)[/tex] increases.
- As [tex]\( x \)[/tex] increases from 1 to 3, [tex]\( g(x) \)[/tex] values approach 1 (since [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{24}{25}\)[/tex] are close to 1).
- Therefore, as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], it appears that [tex]\( g(x) \)[/tex] approaches 1.
Given this analysis, let's compare the statements:
(A) "They have the same x-intercept and the same end behavior as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex]."
- Incorrect because another function must have the same x-intercept (1) and also approach 1 at [tex]\( x \)[/tex] increasing indefinitely.
(B) "They have the same end behavior as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], but they have different x- and y-intercepts."
- This statement can be true if we compare function [tex]\( g(x) \)[/tex] with another function that has the same end behavior (approaching 1), but with different x-intercepts and y-intercepts.
(C) "They have the same x-intercept and the same y-intercept."
- Incorrect because that would imply the functions have identical intercepts, which is not given in the problem.
(D) "They have the same y-intercept and the same end behavior as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex]."
- Incorrect because it is specified that the other function must have the same y-intercept as -4 and also approaches 1 as [tex]\( x \)[/tex] increases indefinitely.
The correct statement is:
(B) They have the same end behavior as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], but they have different x- and y-intercepts.
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