Let's determine whether the given table represents an exponential function.
An exponential function has the form [tex]\( y = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( b \)[/tex] (the base) is the same for each pair of successive [tex]\( x \)[/tex]-values.
Given data in the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 5 \\
\hline
2 & 10 \\
\hline
3 & 15 \\
\hline
4 & 20 \\
\hline
5 & 25 \\
\hline
\end{array}
\][/tex]
To check if this table represents an exponential function, we need to look at the ratios of successive [tex]\( y \)[/tex]-values:
1. Calculate the ratio of [tex]\( y \)[/tex]-values for [tex]\( x = 2 \)[/tex] and [tex]\( x = 1 \)[/tex]:
[tex]\[
\text{Ratio} = \frac{y(2)}{y(1)} = \frac{10}{5} = 2.0
\][/tex]
2. Calculate the ratio of [tex]\( y \)[/tex]-values for [tex]\( x = 3 \)[/tex] and [tex]\( x = 2 \)[/tex]:
[tex]\[
\text{Ratio} = \frac{y(3)}{y(2)} = \frac{15}{10} = 1.5
\][/tex]
3. Calculate the ratio of [tex]\( y \)[/tex]-values for [tex]\( x = 4 \)[/tex] and [tex]\( x = 3 \)[/tex]:
[tex]\[
\text{Ratio} = \frac{y(4)}{y(3)} = \frac{20}{15} \approx 1.333
\][/tex]
4. Calculate the ratio of [tex]\( y \)[/tex]-values for [tex]\( x = 5 \)[/tex] and [tex]\( x = 4 \)[/tex]:
[tex]\[
\text{Ratio} = \frac{y(5)}{y(4)} = \frac{25}{20} = 1.25
\][/tex]
We observe that the ratios between successive [tex]\( y \)[/tex]-values are:
[tex]\[
[2.0, 1.5, 1.333, 1.25]
\][/tex]
Since these ratios are not all the same, we can conclude that the table does not represent an exponential function. For a table to represent an exponential function, the ratio of successive [tex]\( y \)[/tex]-values must be constant.
Therefore, the table does not represent an exponential function.
