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To determine which table of values contains points that lie on the graph of the function [tex]\( f(x) = \sqrt{x} \)[/tex], we need to evaluate the function at each [tex]\( x \)[/tex]-value given in both tables and check which points correspond correctly to the values of [tex]\( f(x) \)[/tex].
### Evaluating [tex]\( f(x) = \sqrt{x} \)[/tex]
For the points in the first table:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -1 & 1 \\ \hline 0 & 0 \\ \hline 1 & 1 \\ \hline 4 & 16 \\ \hline \end{array} \][/tex]
1. [tex]\( f(-1) = \sqrt{-1} \)[/tex]: This results in a complex number [tex]\( (6.123233995736766e-17+1j) \)[/tex], which does not lie on the real graph of [tex]\( \sqrt{x} \)[/tex].
2. [tex]\( f(0) = \sqrt{0} = 0 \)[/tex]: Correct, as [tex]\((0, 0)\)[/tex] lies on the graph of [tex]\( y = \sqrt{x} \)[/tex].
3. [tex]\( f(1) = \sqrt{1} = 1 \)[/tex]: Correct, as [tex]\((1, 1)\)[/tex] lies on the graph of [tex]\( y = \sqrt{x} \)[/tex].
4. [tex]\( f(4) = \sqrt{4} = 2 \)[/tex]: The value [tex]\( f(4) = 16 \)[/tex] given in the table is incorrect. [tex]\((4, 2)\)[/tex] should be the correct point on the graph.
Hence, the first table does not correctly represent the function [tex]\( f(x) = \sqrt{x} \)[/tex].
For the points in the second table:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -1 & 0 \\ \hline 0 & 0 \\ \hline 1 & 1 \\ \hline 4 & 2 \\ \hline \end{array} \][/tex]
1. [tex]\( f(-1) = \sqrt{-1} \)[/tex]: Again, this results in a complex number [tex]\( (6.123233995736766e-17+1j) \)[/tex], which does not lie on the real graph of [tex]\( \sqrt{x} \)[/tex].
2. [tex]\( f(0) = \sqrt{0} = 0 \)[/tex]: Correct, as [tex]\((0, 0)\)[/tex] lies on the graph of [tex]\( y = \sqrt{x} \)[/tex].
3. [tex]\( f(1) = \sqrt{1} = 1 \)[/tex]: Correct, as [tex]\((1, 1)\)[/tex] lies on the graph of [tex]\( y = \sqrt{x} \)[/tex].
4. [tex]\( f(4) = \sqrt{4} = 2 \)[/tex]: Correct, as [tex]\((4, 2)\)[/tex] lies on the graph of [tex]\( y = \sqrt{x} \)[/tex].
Thus, except for the complex number calculated for [tex]\(f(-1)\)[/tex] which is not on the graph, all other points in the second table correctly represent the values of the function [tex]\( f(x) = \sqrt{x} \)[/tex].
### Conclusion
By evaluating [tex]\( f(x) = \sqrt{x} \)[/tex] for the given points, we find that the second table correctly contains points that lie on the graph of the function, except for x = -1. Thus, with this understanding, it is clear that:
- The second table ([tex]\( x \)[/tex] values like [tex]\(0, 1, 4 \)[/tex]) correctly represents the real-valued points on the graph of [tex]\( f(x) \)[/tex].
### Evaluating [tex]\( f(x) = \sqrt{x} \)[/tex]
For the points in the first table:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -1 & 1 \\ \hline 0 & 0 \\ \hline 1 & 1 \\ \hline 4 & 16 \\ \hline \end{array} \][/tex]
1. [tex]\( f(-1) = \sqrt{-1} \)[/tex]: This results in a complex number [tex]\( (6.123233995736766e-17+1j) \)[/tex], which does not lie on the real graph of [tex]\( \sqrt{x} \)[/tex].
2. [tex]\( f(0) = \sqrt{0} = 0 \)[/tex]: Correct, as [tex]\((0, 0)\)[/tex] lies on the graph of [tex]\( y = \sqrt{x} \)[/tex].
3. [tex]\( f(1) = \sqrt{1} = 1 \)[/tex]: Correct, as [tex]\((1, 1)\)[/tex] lies on the graph of [tex]\( y = \sqrt{x} \)[/tex].
4. [tex]\( f(4) = \sqrt{4} = 2 \)[/tex]: The value [tex]\( f(4) = 16 \)[/tex] given in the table is incorrect. [tex]\((4, 2)\)[/tex] should be the correct point on the graph.
Hence, the first table does not correctly represent the function [tex]\( f(x) = \sqrt{x} \)[/tex].
For the points in the second table:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -1 & 0 \\ \hline 0 & 0 \\ \hline 1 & 1 \\ \hline 4 & 2 \\ \hline \end{array} \][/tex]
1. [tex]\( f(-1) = \sqrt{-1} \)[/tex]: Again, this results in a complex number [tex]\( (6.123233995736766e-17+1j) \)[/tex], which does not lie on the real graph of [tex]\( \sqrt{x} \)[/tex].
2. [tex]\( f(0) = \sqrt{0} = 0 \)[/tex]: Correct, as [tex]\((0, 0)\)[/tex] lies on the graph of [tex]\( y = \sqrt{x} \)[/tex].
3. [tex]\( f(1) = \sqrt{1} = 1 \)[/tex]: Correct, as [tex]\((1, 1)\)[/tex] lies on the graph of [tex]\( y = \sqrt{x} \)[/tex].
4. [tex]\( f(4) = \sqrt{4} = 2 \)[/tex]: Correct, as [tex]\((4, 2)\)[/tex] lies on the graph of [tex]\( y = \sqrt{x} \)[/tex].
Thus, except for the complex number calculated for [tex]\(f(-1)\)[/tex] which is not on the graph, all other points in the second table correctly represent the values of the function [tex]\( f(x) = \sqrt{x} \)[/tex].
### Conclusion
By evaluating [tex]\( f(x) = \sqrt{x} \)[/tex] for the given points, we find that the second table correctly contains points that lie on the graph of the function, except for x = -1. Thus, with this understanding, it is clear that:
- The second table ([tex]\( x \)[/tex] values like [tex]\(0, 1, 4 \)[/tex]) correctly represents the real-valued points on the graph of [tex]\( f(x) \)[/tex].
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