To solve this problem, we need to understand the properties of an even function and use the given information about the [tex]$x$[/tex]-intercepts.
An even function is symmetric with respect to the [tex]$y$[/tex]-axis. This means that if \((a, 0)\) is an [tex]$x$[/tex]-intercept, then \((-a, 0)\) must also be an [tex]$x$[/tex]-intercept.
Given that \((6, 0)\) is one of the [tex]$x$[/tex]-intercepts, we must have \((-6, 0)\) as another intercept because \(f(x)\) is even.
We are given several sets of potential [tex]$x$[/tex]-intercepts. We need to determine which of these sets can fit together with \((6, 0)\) and \((-6, 0)\) to make up the five [tex]$x$[/tex]-intercepts.
Let's examine each given option:
1. \( (-6,0), (-2,0), (0,0) \):
- We already know \((-6,0)\) is included.
- \((-2,0)\) is another intercept.
- \((0,0)\) is another intercept.
So, the intercepts in this option are \( (-6,0), (-2,0), (0,0) \). With the additional intercepts \((6, 0)\) and \((-6, 0)\), we have five intercepts: \( (-6,0), (-2,0), (0,0), (6,0), (-6,0) \). This option includes both \(6\) and \(-6\) and makes up the correct number of intercepts.
2. \( (-6,0), (-2,0), (4,0) \):
- We already know \((-6,0)\) is included.
- \((-2,0)\) is another intercept.
- \((4,0)\) is another intercept.
So, the intercepts in this option are \( (-6,0), (-2,0), (4,0) \). With the additional intercepts \((6, 0)\), we get five intercepts: \( (-6,0), (-2,0), (4,0), (6,0), (-6,0) \).
3. \( (-4,0), (0,0), (2,0) \):
- \((-4, 0)\) is one intercept.
- \((0, 0)\) is another intercept.
- \((2, 0)\) is another intercept.
For this to combine with \((6, 0)\) and \((-6, 0)\), we would need \( (-2, 0) \) and \( (0, 0) \), but \((2,0)\) is not the symmetrical intercept of \((6,0)\). Therefore, this option does not fit with the other given intercepts of an even function.
4. \( (-4,0), (-2,0), (0,0) \):
- \((-4,0)\) is one intercept.
- \((-2,0)\) is another intercept.
- \((0,0)\) is another intercept.
Similar to option 3, for this to combine with \((6, 0)\) and \((-6, 0)\), we need symmetric pairs, and the intercepts here do not provide a mirror image with respect to \( \pm 6 \). This option does not fit.
Based on our reasoning, the sets of [tex]$x$[/tex]-intercepts which could be included in the even function graph are:
[tex]\[ \boxed{1 \text{ and } 2} \][/tex]
