Let's analyze the given table and the given statements one by one.
### Table of Values for [tex]\( y = f(x) \)[/tex]
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
x & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 \\
\hline
f(x) & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\
\hline
\end{array}
\][/tex]
### Statements Analysis:
1. [tex]\( f(5) = 6 \)[/tex]:
- Let's check the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 5 \)[/tex], according to the table:
- [tex]\( f(5) = 6 \)[/tex]
- Therefore, this statement is true.
2. 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 input values [tex]\( x \)[/tex]. According to the table, the input values [tex]\( x \)[/tex] are:
- [tex]\( \{5, 10, 15, 20, 25, 30, 35, 40\} \)[/tex]
- The provided domain [tex]\( \{5, 6, 7, 8, 10, 11, 12, 13\} \)[/tex] contains values that are not part of the input values and also misses some input values.
- Therefore, this statement is false.
3. The range for [tex]\( f(x) \)[/tex] is all real numbers:
- The range of a function is the set of all output values [tex]\( f(x) \)[/tex]. According to the table, the output values [tex]\( f(x) \)[/tex] are discrete and are:
- [tex]\( \{6, 7, 8, 9, 10, 11, 12, 13\} \)[/tex]
- This is not the set of all real numbers, but rather a specific set of integer values.
- Therefore, this statement is false.
4. [tex]\( f(15) = 8 \)[/tex]:
- Let's check the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 15 \)[/tex], according to the table:
- [tex]\( f(15) = 8 \)[/tex]
- Therefore, this statement is true.
### Summary:
Given the table and the statements provided, the true and false statements are:
- [tex]\( f(5) = 6 \)[/tex] is true.
- The domain for [tex]\( f(x) \)[/tex] is the set [tex]\( \{5, 6, 7, 8, 10, 11, 12, 13\} \)[/tex] is false.
- The range for [tex]\( f(x) \)[/tex] is all real numbers is false.
- [tex]\( f(15) = 8 \)[/tex] is true.
So, the result is:
[tex]\[ [1, 1, 1, 1] \][/tex]
