Let's examine each statement based on the given table of values for [tex]\( y = f(x) \)[/tex].
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
x & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
f(x) & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\
\hline
\end{array}
\][/tex]
### Statement 1: "The range for [tex]\( f(x) \)[/tex] is all real numbers."
To evaluate this statement, we need to consider the values that [tex]\( f(x) \)[/tex] can take. From the table, the values of [tex]\( f(x) \)[/tex] are:
[tex]\[ f(x) = \{5, 6, 7, 8, 9, 10, 11, 12, 13\} \][/tex]
The range of [tex]\( f(x) \)[/tex] according to the table is a finite set of specific integers. Therefore, the claim that the range is all real numbers is not true because it does not include all possible real numbers—only a specific subset of integers.
So, this statement is false.
### Statement 2: "The domain for [tex]\( f(x) \)[/tex] is the set [tex]\(\{-2, -1, 0, 1, 2, 3, 4, 5, 6\}\)[/tex]."
The domain of a function is the set of all possible input values (x-values) for which the function is defined. From the table, the x-values are:
[tex]\[ x = \{-2, -1, 0, 1, 2, 3, 4, 5, 6\} \][/tex]
This matches exactly with the given set in the statement.
So, this statement is true.
### Statement 3: "[tex]\( f(-1) = 6 \)[/tex]"
To evaluate this statement, look at the value of [tex]\( f(x) \)[/tex] when [tex]\( x = -1 \)[/tex] from the table:
[tex]\[ f(-1) = 6 \][/tex]
Therefore, this statement is true.
### Statement 4: "[tex]\( f(5) = -2 \)[/tex]"
To evaluate this statement, look at the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 5 \)[/tex] from the table:
[tex]\[ f(5) = 12 \][/tex]
Therefore, the statement [tex]\( f(5) = -2 \)[/tex] is not correct, as the actual value is 12.
So, this statement is false.
### Summary
Based on the analysis:
- Statement 1: False
- Statement 2: True
- Statement 3: True
- Statement 4: False
