To solve the problem of adding one number to each column of the table such that it shows a function and does not repeat any ordered pair, follow these steps:
1. Identify the existing pairs and possible new values:
- Existing pairs are:
- (6, 6)
- (3, 8)
- (9, 12)
- (7, 8)
- Possible values to be inserted: 3, 6, 7, 8, 9, 12
2. Find a new pair of (x, y) that keeps the function's definition intact and does not repeat any existing pair:
- A function requires that each input [tex]\( x \)[/tex] has exactly one output [tex]\( y \)[/tex].
3. Check the potential values:
- The potential values (3, 6, 7, 8, 9, 12) have to form a pair (3, y) or (y, 8) where y is any of the potential values.
- Check each possible value to avoid repeating pairs.
4. Form new pairs to ensure no repeating ordered pairs:
- The existing pairs involving '3' are (3, 8) and for '8', the existing pairs are (7, 8) and (3, 8). Therefore, any new pair formed using '3' must not duplicate these pairs.
- Similarly, for the other numbers, we need to ensure they do not already exist in the table and follow the function rule of unique [tex]\( x \)[/tex].
5. Choose the correct value pair:
- Since the pair (3, 8) already exists, and (7, 8) also exists, let's look for another number paired with '3':
- (3, 6)
- (3, 12)
- Similarly, let's avoid any pairs (x, 8) where x is one of the potential numbers left, like:
- (3, 8) ⟶ already exists
- (6, 8) ⟶ not valid as (3, 8) exists
After checking these constraints, the pair (3, 6) is a valid choice.
Here is the detailed solution:
- Add number `3` in the first column under [tex]\( x \)[/tex].
- Add number `6` in the corresponding spot of the second column under [tex]\( y \)[/tex].
The resulting table is:
[tex]\[
\begin{tabular}{|c|c|}
\hline$x$ & $y$ \\
\hline 6 & 6 \\
\hline 3 & 8 \\
\hline 9 & 12 \\
\hline 7 & 8 \\
\hline 3 & 6 \\
\hline
\end{tabular}
\][/tex]
By utilizing the detailed process, the unique pairs formed are:
[tex]\[ (6, 6), (3, 8), (9, 12), (7, 8), \text{and} (3, 6) \][/tex]
Hence, the table successfully shows a function without repeating any ordered pair.
