To determine whether the given table represents a function, we need to understand what constitutes a function. In mathematics, a function is defined as a relation between a set of inputs (called the domain) and a set of possible outputs (called the codomain), where each input is related to exactly one output.
Here is the given table in a structured form:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
3 & 1 \\
\hline
4 & 3 \\
\hline
5 & 2 \\
\hline
6 & 5 \\
\hline
6 & 6 \\
\hline
\end{array}
\][/tex]
Let's analyze this step-by-step:
1. Identify the inputs and outputs:
- Inputs ([tex]$x$[/tex]): 3, 4, 5, 6
- Outputs ([tex]$y$[/tex]): 1, 3, 2, 5, 6
2. Rule for functions: Each input in the domain should correspond to exactly one output in the codomain.
3. Examine each input:
- [tex]$x = 3$[/tex] corresponds to [tex]$y = 1$[/tex]
- [tex]$x = 4$[/tex] corresponds to [tex]$y = 3$[/tex]
- [tex]$x = 5$[/tex] corresponds to [tex]$y = 2$[/tex]
- [tex]$x = 6$[/tex] corresponds to [tex]$y = 5$[/tex] and [tex]$y = 6$[/tex]
4. Check for violations of the function rule:
- For [tex]$x = 6$[/tex], there are two different outputs: [tex]$y = 5$[/tex] and [tex]$y = 6$[/tex]. This violates the definition of a function because a single input ([tex]$x = 6$[/tex]) maps to multiple outputs.
Based on this analysis, we conclude that the given table does not represent a function.
Therefore, the correct answer is:
B. No, because one [tex]$x$[/tex]-value corresponds to two different [tex]$y$[/tex]-values.
