To determine whether [tex]\( f(x) \)[/tex] is a function, we need to check if each input [tex]\( x \)[/tex] in the domain associates with exactly one output [tex]\( y \)[/tex]. In other words, each [tex]\( x \)[/tex] should map to only one [tex]\( y \)[/tex], and there should be no repeated [tex]\( x \)[/tex] values with different [tex]\( y \)[/tex] values.
Here is the sequence of pairs given in the table:
- [tex]\( f(2) = 3 \)[/tex]
- [tex]\( f(3) = 7 \)[/tex]
- [tex]\( f(5) = 12 \)[/tex]
Let's examine the [tex]\( x \)[/tex]-values: [tex]\( 2 \)[/tex], [tex]\( 3 \)[/tex], and [tex]\( 5 \)[/tex].
1. Check [tex]\( f(2) \)[/tex]:
- There is only one [tex]\( y \)[/tex]-value for [tex]\( x = 2 \)[/tex], which is [tex]\( 3 \)[/tex].
2. Check [tex]\( f(3) \)[/tex]:
- There is only one [tex]\( y \)[/tex]-value for [tex]\( x = 3 \)[/tex], which is [tex]\( 7 \)[/tex].
3. Check [tex]\( f(5) \)[/tex]:
- There is only one [tex]\( y \)[/tex]-value for [tex]\( x = 5 \)[/tex], which is [tex]\( 12 \)[/tex].
Since all [tex]\( x \)[/tex]-values map to exactly one [tex]\( y \)[/tex]-value and there are no repeated [tex]\( x \)[/tex]-values paired with different [tex]\( y \)[/tex]-values, we conclude that [tex]\( f(x) \)[/tex] satisfies the definition of a function.
Therefore, the statement that [tex]\( f(x) \)[/tex] is a function is:
○ A. True
