To find the function representing the reflection of [tex]\( f(x) = \sqrt{x} \)[/tex] over the [tex]\( x \)[/tex]-axis, we need to understand what reflection over the [tex]\( x \)[/tex]-axis implies. A reflection over the [tex]\( x \)[/tex]-axis transforms a point [tex]\((x, y)\)[/tex] into [tex]\((x, -y)\)[/tex]. Essentially, you are flipping the graph upside down.
Given the function [tex]\( f(x) = \sqrt{x} \)[/tex], reflecting it over the [tex]\( x \)[/tex]-axis would result in:
[tex]\[ -f(x) = -\sqrt{x} \][/tex]
Now let's verify if this transformation matches the given table values after reflection:
For [tex]\( f(x) = \sqrt{x} \)[/tex]:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = \sqrt{0} = 0 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( y = \sqrt{1} = 1 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = \sqrt{2} \approx 1.41 \)[/tex]
- When [tex]\( x = 4 \)[/tex], [tex]\( y = \sqrt{4} = 2 \)[/tex]
After reflecting these points over the [tex]\( x \)[/tex]-axis:
- When [tex]\( x = 0 \)[/tex], [tex]\( -y = -0 = 0 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( -y = -1 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( -y = -1.41 \)[/tex]
- When [tex]\( x = 4 \)[/tex], [tex]\( -y = -2 \)[/tex]
Let’s compare these values with the values in the second table:
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|}
\hline
[tex]\(x\)[/tex] & [tex]\(y\)[/tex] \\
\hline
0 & 0 \\
\hline
1 & -1 \\
\hline
2 & -1.41 \\
\hline
4 & -2 \\
\hline
\end{tabular}
\end{table}
We see that the [tex]\( y \)[/tex]-values from the reflected function [tex]\( -\sqrt{x} \)[/tex] match the values in this table exactly. Therefore, the function representing the reflection over the [tex]\( x \)[/tex]-axis of [tex]\( f(x) = \sqrt{x} \)[/tex] is:
[tex]\[ g(x) = -\sqrt{x} \][/tex]
