Let's examine the given functions to find out which of them have a vertex at [tex]\( x = 0 \)[/tex].
Firstly, let's understand what it means for a function [tex]\( f(x) \)[/tex] to have a vertex at a particular [tex]\( x \)[/tex] value. For absolute value functions of the form [tex]\( f(x) = |x - h| + k \)[/tex], the vertex is located at [tex]\( x = h \)[/tex].
We are provided with three functions. Let's evaluate each of them at [tex]\( x = 0 \)[/tex]:
1. Function [tex]\( f(x) = |x| \)[/tex]:
- The function [tex]\( f(x) = |x| \)[/tex] has its vertex at [tex]\( x = 0 \)[/tex] because [tex]\( |0| = 0 \)[/tex]. This implies the minimum value is achieved when [tex]\( x = 0 \)[/tex].
2. Function [tex]\( f(x) = |x + 3| \)[/tex]:
- The value of this function at [tex]\( x = 0 \)[/tex] is [tex]\( |0 + 3| = 3 \)[/tex].
- The vertex of the function [tex]\( f(x) = |x + 3| \)[/tex] occurs when [tex]\( x = -3 \)[/tex], not [tex]\( x = 0 \)[/tex].
3. Function [tex]\( f(x) = |x + 3| - 6 \)[/tex]:
- Evaluating this function at [tex]\( x = 0 \)[/tex], we get [tex]\( |0 + 3| - 6 = 3 - 6 = -3 \)[/tex].
- The vertex of this function, [tex]\( f(x) = |x + 3| - 6 \)[/tex], also occurs at [tex]\( x = -3 \)[/tex], not [tex]\( x = 0 \)[/tex].
Based on these evaluations:
- [tex]\( f(x) = |x| \)[/tex] achieves the minimum value at [tex]\( x = 0 \)[/tex]. So, it has a vertex at [tex]\( x = 0 \)[/tex].
- [tex]\( f(x) = |x + 3| \)[/tex] does not have a vertex at [tex]\( x = 0 \)[/tex].
- [tex]\( f(x) = |x + 3| - 6 \)[/tex] does not have a vertex at [tex]\( x = 0 \)[/tex].
Thus, only the function [tex]\( f(x) = |x| \)[/tex] has a vertex at [tex]\( x = 0 \)[/tex].
The correct option with a vertex at [tex]\( x = 0 \)[/tex] is:
[tex]\[ \boxed{1} \][/tex]
So, the function [tex]\( f(x) = |x| \)[/tex] is the only one that has a vertex with an [tex]\( x \)[/tex]-value of 0.
