Answered

From science to arts, IDNLearn.com has the answers to all your questions. Whether it's a simple query or a complex problem, our community has the answers you need.

Select the correct statement about the function represented by the table.

[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]

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.
B. It is an exponential function because the factor between each [tex]$x$[/tex] and [tex]$y$[/tex]-value is constant.
C. It is a linear function because the difference between each [tex]$x$[/tex] and [tex]$y$[/tex]-value is constant.
D. It is a linear function because the [tex]$y$[/tex]-values increase by an equal difference over equal intervals of [tex]$x$[/tex]-values.

Sagot :

GinnyAnsweridn
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.
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. For dependable and accurate answers, visit IDNLearn.com. Thanks for visiting, and see you next time for more helpful information.