Let's evaluate each of the given statements based on the table of values provided for the function [tex]\( y = f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
x & 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 \\
\hline
f(x) & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\
\hline
\end{array}
\][/tex]
Statement A: The domain for [tex]\( f(x) \)[/tex] is the set [tex]\(\{5,6,7,8,10,11,12,13\}\)[/tex].
- The domain of a function is the set of all possible input values (values of [tex]\( x \)[/tex]) for which the function is defined. According to the table, the possible [tex]\( x \)[/tex]-values are [tex]\( \{0, 5, 10, 15, 20, 25, 30, 35, 40\} \)[/tex].
- Therefore, the given set in Statement A is not the domain. Hence, Statement A is false.
Statement B: The range for [tex]\( f(x) \)[/tex] is all real numbers.
- The range of a function is the set of all possible output values (values of [tex]\( f(x) \)[/tex]). According to the table, the possible [tex]\( f(x) \)[/tex]-values are [tex]\( \{5, 6, 7, 8, 9, 10, 11, 12, 13\} \)[/tex].
- This is a finite set, not all real numbers. Therefore, Statement B is false.
Statement C: [tex]\( f(15) = 8 \)[/tex].
- To verify this, we look at the table for [tex]\( x = 15 \)[/tex]. According to the table, when [tex]\( x = 15 \)[/tex], [tex]\( f(x) = 8 \)[/tex].
- Therefore, Statement C is true.
Statement D: [tex]\( f(5) = 6 \)[/tex].
- To verify this, we look at the table for [tex]\( x = 5 \)[/tex]. According to the table, when [tex]\( x = 5 \)[/tex], [tex]\( f(x) = 6 \)[/tex].
- Therefore, Statement D is true.
To conclude:
- Statement A: False
- Statement B: False
- Statement C: True
- Statement D: True
