To identify the reflection of [tex]\( f(x) = \sqrt{x} \)[/tex] over the [tex]\( x \)[/tex]-axis, it's essential to determine the reflected points of [tex]\( f(x) \)[/tex].
Recall that reflecting a function [tex]\( f(x) \)[/tex] over the [tex]\( x \)[/tex]-axis transforms [tex]\( f(x) \)[/tex] to [tex]\( -f(x) \)[/tex]. This means that for every point [tex]\( (x, f(x)) \)[/tex] on the graph of [tex]\( f(x) \)[/tex], the corresponding point on the reflected graph is [tex]\( (x, -f(x)) \)[/tex].
Given the values of [tex]\( f(x) \)[/tex] in the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-1 & \text{undefined} \\
\hline
0 & 0 \\
\hline
1 & 1 \\
\hline
4 & 2 \\
\hline
\end{array}
\][/tex]
We reflect each defined point over the [tex]\( x \)[/tex]-axis:
1. For [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 0 \)[/tex]. Its reflection is [tex]\( (0, -0) = (0, 0) \)[/tex].
2. For [tex]\( x = 1 \)[/tex], [tex]\( f(1) = 1 \)[/tex]. Its reflection is [tex]\( (1, -1) \)[/tex].
3. For [tex]\( x = 4 \)[/tex], [tex]\( f(4) = 2 \)[/tex]. Its reflection is [tex]\( (4, -2) \)[/tex].
Summarizing these reflections:
[tex]\[
\begin{array}{|c|c|}
\hline
x & -f(x) \\
\hline
0 & 0 \\
\hline
1 & -1 \\
\hline
4 & -2 \\
\hline
\end{array}
\][/tex]
Therefore, the reflected values of [tex]\( f(x) = \sqrt{x} \)[/tex] over the [tex]\( x \)[/tex]-axis are:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y = -f(x) \\
\hline
0 & 0 \\
\hline
1 & -1 \\
\hline
4 & -2 \\
\hline
\end{array}
\][/tex]
These reflect the points correctly as [tex]\( (0, 0) \)[/tex], [tex]\( (1, -1) \)[/tex], and [tex]\( (4, -2) \)[/tex].
So, the correct representation of the reflection over the [tex]\( x \)[/tex]-axis in the given tabular format would be:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y = -f(x) \\
\hline
-1 & \text{undefined} \\
\hline
0 & 0 \\
\hline
1 & -1 \\
\hline
4 & -2 \\
\hline
\end{array}
\][/tex]
These values match the result for the reflected points as calculated.
