To determine the correct statement about the function represented by the table, we need to analyze how the [tex]\( y \)[/tex]-values change relative to the [tex]\( x \)[/tex]-values.
First, let's look at the given data:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 7 \\
\hline
2 & 21 \\
\hline
3 & 63 \\
\hline
4 & 189 \\
\hline
5 & 567 \\
\hline
\end{array}
\][/tex]
Our goal is to determine whether the function is exponential or linear and find the correct reason.
#### Exponential Function Check
An exponential function has the property that the ratio between consecutive [tex]\( y \)[/tex]-values is constant. Let's calculate the ratio between consecutive [tex]\( y \)[/tex]-values:
- For [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
[tex]\[
\frac{21}{7} = 3
\][/tex]
- For [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex]:
[tex]\[
\frac{63}{21} = 3
\][/tex]
- For [tex]\( x = 3 \)[/tex] and [tex]\( x = 4 \)[/tex]:
[tex]\[
\frac{189}{63} = 3
\][/tex]
- For [tex]\( x = 4 \)[/tex] and [tex]\( x = 5 \)[/tex]:
[tex]\[
\frac{567}{189} = 3
\][/tex]
Since the ratio [tex]\( \frac{y_{i+1}}{y_i} \)[/tex] is consistently 3 for all consecutive pairs of [tex]\( y \)[/tex]-values, the function is exponential.
#### Linear Function Check
A linear function has the property that the difference between consecutive [tex]\( y \)[/tex]-values is constant. Let's calculate the difference between consecutive [tex]\( y \)[/tex]-values:
- For [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
[tex]\[
21 - 7 = 14
\][/tex]
- For [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex]:
[tex]\[
63 - 21 = 42
\][/tex]
- For [tex]\( x = 3 \)[/tex] and [tex]\( x = 4 \)[/tex]:
[tex]\[
189 - 63 = 126
\][/tex]
- For [tex]\( x = 4 \)[/tex] and [tex]\( x = 5 \)[/tex]:
[tex]\[
567 - 189 = 378
\][/tex]
Since the differences [tex]\( y_{i+1} - y_i \)[/tex] are not constant, the function is not linear.
Based on this analysis, we can determine the correct statement. The [tex]\( y \)[/tex]-values increase by an equal factor (3) over equal intervals of [tex]\( x \)[/tex]-values, which is a characteristic of an exponential function.
Hence, the correct answer is:
A. It is an exponential function because the [tex]\( y \)[/tex]-values increase by an equal factor over equal intervals of [tex]\( x \)[/tex]-values.
