To determine for which values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] the given table represents a function, we need to ensure that every [tex]\(x\)[/tex]-value in the table is unique. A function must allocate only one [tex]\(y\)[/tex]-value for each [tex]\(x\)[/tex]-value, thus the [tex]\(x\)[/tex]-values must not repeat.
The given options to test for [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
1. [tex]\(a=0\)[/tex] and [tex]\(b=0\)[/tex]
2. [tex]\(a=5\)[/tex] and [tex]\(b=5\)[/tex]
3. [tex]\(a=4\)[/tex] and [tex]\(b=7\)[/tex]
4. [tex]\(a=3\)[/tex] and [tex]\(b=1\)[/tex]
Let's analyze each case:
### Case 1: [tex]\(a=0\)[/tex] and [tex]\(b=0\)[/tex]
Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 9 \\
\hline
3 & 4 \\
\hline
5 & 0 \\
\hline
7 & 2 \\
\hline
0 & 0 \\
\hline
\end{array}
\][/tex]
The [tex]\(x\)[/tex]-values are \{1, 3, 5, 7, 0\}, which are all unique.
### Case 2: [tex]\(a=5\)[/tex] and [tex]\(b=5\)[/tex]
Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 9 \\
\hline
3 & 4 \\
\hline
5 & 5 \\
\hline
7 & 2 \\
\hline
5 & 0 \\
\hline
\end{array}
\][/tex]
The [tex]\(x\)[/tex]-values are \{1, 3, 5, 7, 5\}, with 5 appearing twice. Thus, this does not represent a function.
### Case 3: [tex]\(a=4\)[/tex] and [tex]\(b=7\)[/tex]
Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 9 \\
\hline
3 & 4 \\
\hline
5 & 4 \\
\hline
7 & 2 \\
\hline
7 & 0 \\
\hline
\end{array}
\][/tex]
The [tex]\(x\)[/tex]-values are \{1, 3, 5, 7, 7\}, with 7 appearing twice. Thus, this does not represent a function.
### Case 4: [tex]\(a=3\)[/tex] and [tex]\(b=1\)[/tex]
Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 9 \\
\hline
3 & 4 \\
\hline
5 & 3 \\
\hline
7 & 2 \\
\hline
1 & 0 \\
\hline
\end{array}
\][/tex]
The [tex]\(x\)[/tex]-values are \{1, 3, 5, 7, 1\}, with 1 appearing twice. Thus, this does not represent a function.
### Conclusion
After evaluating all cases, the only combination that keeps all [tex]\(x\)[/tex]-values unique is:
[tex]\[ a = 0 \text{ and } b = 0 \][/tex]
Thus, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that make the table represent a function are [tex]\(\boxed{a = 0 \text{ and } b = 0}\)[/tex].
