To determine the type of function represented by the given table, we can analyze the differences between the consecutive [tex]\( y \)[/tex]-values (cost in dollars).
The table given is:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
y & 15 & 25 & 35 & 45 & 55 & 65 \\
\hline
\end{array}
\][/tex]
We need to find the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[
\begin{align*}
y_2 - y_1 &= 25 - 15 = 10, \\
y_3 - y_2 &= 35 - 25 = 10, \\
y_4 - y_3 &= 45 - 35 = 10, \\
y_5 - y_4 &= 55 - 45 = 10, \\
y_6 - y_5 &= 65 - 55 = 10.
\end{align*}
\][/tex]
The differences between each consecutive pair of [tex]\( y \)[/tex]-values are all equal to 10. This consistent difference indicates that the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is characterized by a constant rate of change.
A function that has a constant rate of change is called a linear function. The form of a linear function is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope (rate of change).
Since the differences between [tex]\( y \)[/tex]-values are consistent and identical, we conclude that the data in the table represents a linear function. Therefore, the type of function represented is:
linear
