Answered

Find solutions to your problems with the help of IDNLearn.com's knowledgeable users. Get the information you need from our community of experts who provide accurate and thorough answers to all your questions.

Which table represents an exponential function of the form [tex]\( y = b^x \)[/tex] when [tex]\( 0 \ \textless \ b \ \textless \ 1 \)[/tex]?

[tex]\[
\begin{tabular}{|c|c|}
\hline
x & y \\
\hline
-3 & \frac{1}{27} \\
\hline
-2 & \frac{1}{9} \\
\hline
-1 & \frac{1}{3} \\
\hline
0 & 1 \\
\hline
1 & 3 \\
\hline
2 & 9 \\
\hline
3 & 27 \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{tabular}{|c|c|}
\hline
x & y \\
\hline
-3 & 27 \\
\hline
-2 & 9 \\
\hline
-1 & 3 \\
\hline
0 & 1 \\
\hline
1 & \frac{1}{3} \\
\hline
2 & \frac{1}{9} \\
\hline
3 & \frac{1}{27} \\
\hline
\end{tabular}
\][/tex]

Sagot :

GinnyAnsweridn
To determine which table represents an exponential function of the form [tex]\( y = b^x \)[/tex] with [tex]\( 0 < b < 1 \)[/tex], let's analyze each table carefully.

### First Table Analysis:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & \frac{1}{27} \\ \hline -2 & \frac{1}{9} \\ \hline -2 & \frac{1}{3} \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 9 \\ \hline 3 & 27 \\ \hline \end{array} \][/tex]

- For [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]: [tex]\( \frac{1/9}{1/27} = 3 \)[/tex]
- For [tex]\( x = -2 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\( \frac{1}{1/9} = 9 \)[/tex]
- For [tex]\( x = 0 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( \frac{9}{1} = 9 \)[/tex]
- For [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\( \frac{27}{9} = 3 \)[/tex]

We can see that the values don't maintain a consistent ratio, suggesting it is not an exponential function of the desired form.

### Second Table Analysis:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & 27 \\ \hline -2 & 9 \\ \hline -1 & 3 \\ \hline 0 & 1 \\ \hline 1 & \frac{1}{3} \\ \hline 2 & \frac{1}{9} \\ \hline 3 & \frac{1}{27} \\ \hline \end{array} \][/tex]

- For [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]: [tex]\( \frac{9}{27} = \frac{1}{3} \)[/tex]
- For [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex]: [tex]\( \frac{3}{9} = \frac{1}{3} \)[/tex]
- For [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\( \frac{1}{3} = \frac{1}{3} \)[/tex]
- For [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\( \frac{1/3}{1} = \frac{1}{3} \)[/tex]
- For [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( \frac{1/9}{1/3} = \frac{1}{3} \)[/tex]
- For [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\( \frac{1/27}{1/9} = \frac{1}{3} \)[/tex]

Each ratio [tex]\( \frac{y_{n+1}}{y_{n}} \)[/tex] is [tex]\( \frac{1}{3} \)[/tex], which is consistent and within the range [tex]\( 0 < b < 1 \)[/tex].

Therefore, the second table represents an exponential function of the form [tex]\( y = b^x \)[/tex] with [tex]\( 0 < b < 1 \)[/tex].

The table number is [tex]\( \boxed{2} \)[/tex].
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For clear and precise answers, choose IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.