Let's evaluate each statement given the 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 A: [tex]\( f(5) = -2 \)[/tex]
We look at the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 5 \)[/tex]. According to the table:
[tex]\[
f(5) = 12
\][/tex]
Since [tex]\( f(5) = 12 \)[/tex] and not [tex]\(-2\)[/tex], this statement is false.
Statement B: [tex]\( f(-1) = 6 \)[/tex]
We look at the value of [tex]\( f(x) \)[/tex] when [tex]\( x = -1 \)[/tex]. According to the table:
[tex]\[
f(-1) = 6
\][/tex]
Since [tex]\( f(-1) = 6 \)[/tex], this statement is true.
Statement C: The domain for [tex]\( f(x) \)[/tex] is the set [tex]\(\{-2, -1, 0, 1, 2, 3, 4, 5, 6\}\)[/tex]
The domain of [tex]\( f(x) \)[/tex] includes all the [tex]\( x \)[/tex]-values present in the table. From the table, the domain is:
[tex]\[
\{-2, -1, 0, 1, 2, 3, 4, 5, 6\}
\][/tex]
Since this matches the given set, this statement is true.
Statement D: The range for [tex]\( f(x) \)[/tex] is all real numbers
The range of [tex]\( f(x) \)[/tex] includes all the [tex]\( y \)[/tex]-values generated by [tex]\( f(x) \)[/tex] in the table. From the table, the range is:
[tex]\[
\{5, 6, 7, 8, 9, 10, 11, 12, 13\}
\][/tex]
The range is a specific set of numbers, not all real numbers. Therefore, this statement is false.
Summary:
- Statement A is false.
- Statement B is true.
- Statement C is true.
- Statement D is false.
