To complete the given table and determine which graph corresponds to the function [tex]\( f(x) = 2 \sqrt{x} \)[/tex], let’s follow a step-by-step procedure:
1. Evaluate [tex]\( f(x) \)[/tex] for [tex]\( x = 0 \)[/tex]:
- Plug [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) = 2 \sqrt{x} \)[/tex].
- [tex]\[ f(0) = 2 \sqrt{0} = 2 \times 0 = 0 \][/tex]
2. Evaluate [tex]\( f(x) \)[/tex] for [tex]\( x = 1 \)[/tex]:
- Plug [tex]\( x = 1 \)[/tex] into the function [tex]\( f(x) = 2 \sqrt{x} \)[/tex].
- [tex]\[ f(1) = 2 \sqrt{1} = 2 \times 1 = 2 \][/tex]
3. Evaluate [tex]\( f(x) \)[/tex] for [tex]\( x = 4 \)[/tex]:
- Plug [tex]\( x = 4 \)[/tex] into the function [tex]\( f(x) = 2 \sqrt{x} \)[/tex].
- [tex]\[ f(4) = 2 \sqrt{4} = 2 \times 2 = 4 \][/tex]
Based on the evaluations, the completed table is:
[tex]\[
\begin{tabular}{|l|l|}
\hline
$x$ & $f(x)$ \\
\hline
0 & 0.0 \\
\hline
1 & 2.0 \\
\hline
4 & 4.0 \\
\hline
\end{tabular}
\][/tex]
Now, let’s interpret this result:
- When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0.0 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( f(x) = 2.0 \)[/tex]
- When [tex]\( x = 4 \)[/tex], [tex]\( f(x) = 4.0 \)[/tex]
To graph the function [tex]\( f(x) = 2 \sqrt{x} \)[/tex], you would plot these points:
- [tex]\( (0, 0.0) \)[/tex]
- [tex]\( (1, 2.0) \)[/tex]
- [tex]\( (4, 4.0) \)[/tex]
Connecting these points with a smooth curve, you should observe a graph starting from the origin (0,0), rising gradually to the point (1,2), and continuing to rise to the point (4,4). The graph is typically in the shape of a gentle curve increasing steadily because the square root function grows slower initially and then accelerates as x increases, but the multiplication by 2 stretches it vertically.
Thus, by plotting these points and connecting them smoothly, you will get the graph of [tex]\( f(x) = 2 \sqrt{x} \)[/tex].
