To find the function that represents the reflection over the [tex]\(x\)[/tex]-axis of the function [tex]\( f(x) = \sqrt{x} \)[/tex], we perform the following steps:
1. Understand Vertical Reflections: Reflecting a function over the [tex]\( x \)[/tex]-axis means that for any point [tex]\((x, y)\)[/tex] on the graph of the original function, there is a corresponding point [tex]\((x, -y)\)[/tex] on the graph of the reflected function. In other words, every [tex]\( y \)[/tex]-value of the function [tex]\( f(x) \)[/tex] gets negated. This can be achieved by multiplying the original function by [tex]\(-1\)[/tex].
2. Apply Reflection to Function:
Since our original function is [tex]\( f(x) = \sqrt{x} \)[/tex], the reflected function over the [tex]\( x \)[/tex]-axis is given by:
[tex]\[
g(x) = -\sqrt{x}
\][/tex]
Hence, the function that represents the reflection over the [tex]\( x \)[/tex]-axis of [tex]\( f(x) = \sqrt{x} \)[/tex] is [tex]\( g(x) = -\sqrt{x} \)[/tex].
The table of values for [tex]\( g(x) \)[/tex] will look like this:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 0 \\
\hline
1 & -1 \\
\hline
2 & -\sqrt{2} \approx -1.41 \\
\hline
4 & -2 \\
\hline
\end{array}
\][/tex]
So, the corresponding reflection over the [tex]\( x \)[/tex]-axis is [tex]\( g(x) = -\sqrt{x} \)[/tex].
