Let's analyze the given data table and determine the true statements about the function [tex]\( g(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
-2 & 2 \\
\hline
-1 & -3 \\
\hline
0 & 2 \\
\hline
1 & 17 \\
\hline
\end{array}
\][/tex]
We are asked to evaluate the following statements:
1. The function is decreasing from [tex]\( x = -2 \)[/tex] to [tex]\( x = 1 \)[/tex].
2. The function is decreasing from [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex].
3. The function is increasing from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex].
4. The function is decreasing from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex].
Let's evaluate each statement in detail:
### Statement 1
The function is decreasing from [tex]\( x = -2 \)[/tex] to [tex]\( x = 1 \)[/tex].
For this statement to be true, [tex]\( g(x) \)[/tex] must continuously decrease as [tex]\( x \)[/tex] increases from [tex]\( -2 \)[/tex] to [tex]\( 1 \)[/tex].
Evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = -2, -1, 0, \)[/tex] and [tex]\( 1 \)[/tex]:
[tex]\[ g(-2) = 2 \][/tex]
[tex]\[ g(-1) = -3 \][/tex]
[tex]\[ g(0) = 2 \][/tex]
[tex]\[ g(1) = 17 \][/tex]
From [tex]\( x=-2 \)[/tex] to [tex]\( x=-1 \)[/tex], [tex]\( g(x) \)[/tex] decreases from 2 to -3.
From [tex]\( x=-1 \)[/tex] to [tex]\( x=0 \)[/tex], [tex]\( g(x) \)[/tex] increases from -3 to 2.
From [tex]\( x=0 \)[/tex] to [tex]\( x=1 \)[/tex], [tex]\( g(x) \)[/tex] increases from 2 to 17.
Since [tex]\( g(x) \)[/tex] increases for some intervals within [tex]\( x = -2 \)[/tex] to [tex]\( x = 1 \)[/tex], the function is not continuously decreasing throughout this range.
Statement 1 is false.
### Statement 2
The function is decreasing from [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex].
Evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex]:
[tex]\[ g(-1) = -3 \][/tex]
[tex]\[ g(0) = 2 \][/tex]
From [tex]\( x=-1 \)[/tex] to [tex]\( x=0 \)[/tex], the function increases from -3 to 2.
Statement 2 is false.
### Statement 3
The function is increasing from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex].
Evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex]:
[tex]\[ g(0) = 2 \][/tex]
[tex]\[ g(1) = 17 \][/tex]
From [tex]\( x=0 \)[/tex] to [tex]\( x=1 \)[/tex], the function increases from 2 to 17.
Statement 3 is true.
### Statement 4
The function is decreasing from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex].
Evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex]:
[tex]\[ g(0) = 2 \][/tex]
[tex]\[ g(1) = 17 \][/tex]
From [tex]\( x=0 \)[/tex] to [tex]\( x=1 \)[/tex], the function increases from 2 to 17, not decreases.
Statement 4 is false.
Thus, the only true statement is:
The function is increasing from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex].
