To analyze the graph representing the total shipping cost [tex]\( y \)[/tex] in dollars for items weighing [tex]\( x \)[/tex] ounces, let's examine the given information and statements.
### Given Table:
[tex]\[
\begin{tabular}{|c|c|}
\hline \text{Weight, in oz} & \text{Total Shipping Cost} \\
\hline \text{not more than 5} & \$ 9.50 \\
\hline \text{more than 5, not more than 10} & \$ 13.25 \\
\hline \text{more than 10, not more than 15} & \$ 17.00 \\
\hline \text{more than 15, not more than 20} & \$ 20.75 \\
\hline
\end{tabular}
\][/tex]
### Statements to Evaluate:
1. The graph is discontinuous at [tex]\( x = 5 \)[/tex], [tex]\( x = 10 \)[/tex], and [tex]\( x = 15 \)[/tex].
2. The range values for the set of the data are [tex]\( \$10, \$14, \$17 \)[/tex], and [tex]\( \$21 \)[/tex].
3. For every 1-ounce increase in weight, the total shipping cost increases by [tex]\( \$3.75 \)[/tex].
### Analysis:
1. Discontinuities:
- A graph is discontinuous at points where there is an abrupt change or jump in the value of the dependent variable.
- From the table, we observe jumps in the shipping cost at:
- [tex]\( x = 5 \)[/tex] (from \[tex]$9.50 to \$[/tex]13.25)
- [tex]\( x = 10 \)[/tex] (from \[tex]$13.25 to \$[/tex]17.00)
- [tex]\( x = 15 \)[/tex] (from \[tex]$17.00 to \$[/tex]20.75)
- Therefore, the graph is indeed discontinuous at [tex]\( x = 5 \)[/tex], [tex]\( x = 10 \)[/tex], and [tex]\( x = 15 \)[/tex].
2. Range Values:
- The range values correspond to the different total shipping costs given in the table.
- According to the table, the shipping costs are:
- \[tex]$9.50
- \$[/tex]13.25
- \[tex]$17.00
- \$[/tex]20.75
- The statement claiming the range values are [tex]\( \$10, \$14, \$17 \)[/tex], and [tex]\( \$21 \)[/tex] is incorrect.
3. Cost Increase per Ounce:
- To determine if the cost increases by \[tex]$3.75 for each 1-ounce increment, we must examine the intervals:
- From not more than 5 oz to more than 5 oz but not more than 10 oz: Increase is \( \$[/tex]13.25 - \[tex]$9.50 = \$[/tex]3.75 \), but this increase does not occur with each ounce.
- From more than 10 oz to more than 15 oz: Increase is again [tex]\( \$17.00 - \$13.25 = \$3.75 \)[/tex]
- However, the cost does not increase by exactly \[tex]$3.75 for every single ounce increment; it only increases at specific weight intervals.
- Thus, the statement claiming a consistent increase of \$[/tex]3.75 for every 1-ounce increment is incorrect.
### Conclusion:
- True Statement: The graph is discontinuous at [tex]\( x = 5 \)[/tex], [tex]\( x = 10 \)[/tex], and [tex]\( x = 15 \)[/tex].
- Incorrect Statements:
- The range values are [tex]\( \$9.50, \$13.25, \$17.00 \)[/tex], and [tex]\( \$20.75 \)[/tex], not [tex]\( \$10, \$14, \$17 \)[/tex], and [tex]\( \$21 \)[/tex].
- The total shipping cost does not uniformly increase by \$3.75 for every single ounce increase.
Thus, the correct evaluation and detailed reasoning confirm the true statement regarding the discontinuities at [tex]\( x = 5 \)[/tex], [tex]\( x = 10 \)[/tex], and [tex]\( x = 15 \)[/tex].
