Let's analyze each statement based on the given table of values for [tex]\( f(x) \)[/tex].
### Statement A: The domain for [tex]\( f(x) \)[/tex] is the set [tex]\(\{5, 6, 7, 8, 9, 10, 11, 12, 13\}\)[/tex].
The domain of a function consists of all the input values (or [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]. These input values are different from the set given in statement A.
So, Statement A is false.
### Statement B: The range for [tex]\( f(x) \)[/tex] is all real numbers.
The range of a function consists of all the output values (or [tex]\( f(x) \)[/tex]-values) that the function can take. From the table, the [tex]\( f(x) \)[/tex]-values are [tex]\(\{5, 6, 7, 8, 9, 10, 11, 12, 13\}\)[/tex]. These are specific integers and do not cover all real numbers.
So, Statement B is false.
### Statement C: [tex]\( f(5) = 6 \)[/tex]
To verify this statement, we look at the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 5 \)[/tex] from the table. The corresponding [tex]\( f(x) \)[/tex] value is 6.
So, Statement C is true.
### Statement D: [tex]\( f(15) = 8 \)[/tex]
To verify this statement, we look at the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 15 \)[/tex] from the table. The corresponding [tex]\( f(x) \)[/tex] value is 8.
So, Statement D is true.
The correct marking for the statements are:
- A: False
- B: False
- C: True
- D: True
