Let's examine each of the statements in detail based on the given table of values for the function [tex]\( y = f(x) \)[/tex]:
| [tex]\( x \)[/tex] | 0 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 |
|---|----|----|----|----|----|----|----|----|----|
| [tex]\( f(x) \)[/tex] | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
### 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 ( [tex]\(x\)[/tex] values) for which the function is defined.
- From the table, the [tex]\(x\)[/tex] values are [tex]\(\{0, 5, 10, 15, 20, 25, 30, 35, 40\}\)[/tex].
- The statement claims that the domain is [tex]\(\{5, 6, 7, 8, 10, 11, 12, 13\}\)[/tex], which includes values that are actually the function's outputs, not inputs.
- Therefore, this statement is False.
### Statement B
[tex]\(f(5) = 6\)[/tex]
- This statement is asking about the specific function value when [tex]\(x = 5\)[/tex].
- According to the table, when [tex]\(x = 5\)[/tex], [tex]\(f(x) = 6\)[/tex].
- Therefore, this statement is True.
### Statement C
The range for [tex]\( f(x) \)[/tex] is all real numbers.
- The range of a function is the set of all possible output values ( [tex]\(f(x)\)[/tex] values).
- From the table, the [tex]\(f(x)\)[/tex] values (range) are [tex]\(\{5, 6, 7, 8, 9, 10, 11, 12, 13\}\)[/tex].
- This set of values is not all real numbers; it is a specific set of integers from 5 to 13.
- Therefore, this statement is False.
### Statement D
[tex]\( f(15) = q \)[/tex]
- This statement expresses the function's output at [tex]\(x = 15\)[/tex] and equates it to a variable [tex]\(q\)[/tex].
- According to the table, [tex]\(f(15) = 8\)[/tex].
- Since [tex]\(q\)[/tex] is not given a value, we compare it to 8.
- However, without further context, the verification of this equivalency would result in us assuming [tex]\(q \neq 8\)[/tex].
- Therefore, this statement is False.
### Summary
Based on the table and the definitions for domain and range:
- A. False
- B. True
- C. False
- D. False
Thus, the correct assessments of the statements are: [tex]\((\text{False}, \text{True}, \text{False}, \text{False})\)[/tex].
