Answer :
Given two tables of values, we are to determine which table represents the exponential decay function of the form [tex]\( y = b^x \)[/tex] where [tex]\( 0 < b < 1 \)[/tex].
Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & \frac{1}{27} \\ \hline -2 & \frac{1}{9} \\ \hline -1 & \frac{1}{3} \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 9 \\ \hline 3 & 27 \\ \hline \end{array} \][/tex]
Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & 27 \\ \hline -2 & 9 \\ \hline -1 & 3 \\ \hline 0 & 1 \\ \hline 1 & \frac{1}{3} \\ \hline 2 & \frac{1}{9} \\ \hline \end{array} \][/tex]
To identify which table represents an exponential decay function [tex]\( y = b^x \)[/tex] where [tex]\( 0 < b < 1 \)[/tex], we need to check the values of [tex]\( y \)[/tex] for the given range of [tex]\( x \)[/tex].
Table 1 Analysis:
For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 1 \][/tex]
This corresponds to [tex]\( b^0 = 1 \)[/tex], which is consistent as [tex]\( b^0 = 1 \)[/tex] for any [tex]\( b \)[/tex].
For positive values of [tex]\( x \)[/tex] (e.g., [tex]\( x = 1 \)[/tex]):
[tex]\[ y = 3 \][/tex]
If [tex]\( y = b \)[/tex] when [tex]\( x = 1 \)[/tex], then [tex]\( b = 3 \)[/tex]. This is not in the range [tex]\( 0 < b < 1 \)[/tex].
Since [tex]\( y \)[/tex] values increase as [tex]\( x \)[/tex] increases and exceed 1, it indicates [tex]\( b > 1 \)[/tex] which does not satisfy [tex]\( 0 < b < 1 \)[/tex].
Therefore, Table 1 does not represent an exponential decay function where [tex]\( 0 < b < 1 \)[/tex].
Table 2 Analysis:
For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 1 \][/tex]
This corresponds to [tex]\( b^0 = 1 \)[/tex], which is consistent.
For negative values of [tex]\( x \)[/tex] (e.g., [tex]\( x = -1 \)[/tex]):
[tex]\[ y = 3 \][/tex]
If [tex]\( y = b^{-1} \)[/tex] when [tex]\( x = -1 \)[/tex], then [tex]\( y = \frac{1}{b} = 3 \)[/tex], thus [tex]\( b = \frac{1}{3} \)[/tex].
If [tex]\( b = \frac{1}{3} \)[/tex]:
For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = \left(\frac{1}{3}\right)^1 = \frac{1}{3} \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \][/tex]
For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = \left(\frac{1}{3}\right)^{-2} = 9 \][/tex]
For [tex]\( x = -3 \)[/tex]:
[tex]\[ y = \left(\frac{1}{3}\right)^{-3} = 27 \][/tex]
These values match exactly with those in Table 2.
Thus, Table 2 correctly represents an exponential function of the form [tex]\( y = b^x \)[/tex] where [tex]\( b = \frac{1}{3} \)[/tex] and [tex]\( 0 < b < 1 \)[/tex].
Therefore, the table that represents the exponential decay function [tex]\( y = b^x \)[/tex] with [tex]\( 0 < b < 1 \)[/tex] is:
Table 2
Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & \frac{1}{27} \\ \hline -2 & \frac{1}{9} \\ \hline -1 & \frac{1}{3} \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 9 \\ \hline 3 & 27 \\ \hline \end{array} \][/tex]
Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & 27 \\ \hline -2 & 9 \\ \hline -1 & 3 \\ \hline 0 & 1 \\ \hline 1 & \frac{1}{3} \\ \hline 2 & \frac{1}{9} \\ \hline \end{array} \][/tex]
To identify which table represents an exponential decay function [tex]\( y = b^x \)[/tex] where [tex]\( 0 < b < 1 \)[/tex], we need to check the values of [tex]\( y \)[/tex] for the given range of [tex]\( x \)[/tex].
Table 1 Analysis:
For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 1 \][/tex]
This corresponds to [tex]\( b^0 = 1 \)[/tex], which is consistent as [tex]\( b^0 = 1 \)[/tex] for any [tex]\( b \)[/tex].
For positive values of [tex]\( x \)[/tex] (e.g., [tex]\( x = 1 \)[/tex]):
[tex]\[ y = 3 \][/tex]
If [tex]\( y = b \)[/tex] when [tex]\( x = 1 \)[/tex], then [tex]\( b = 3 \)[/tex]. This is not in the range [tex]\( 0 < b < 1 \)[/tex].
Since [tex]\( y \)[/tex] values increase as [tex]\( x \)[/tex] increases and exceed 1, it indicates [tex]\( b > 1 \)[/tex] which does not satisfy [tex]\( 0 < b < 1 \)[/tex].
Therefore, Table 1 does not represent an exponential decay function where [tex]\( 0 < b < 1 \)[/tex].
Table 2 Analysis:
For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 1 \][/tex]
This corresponds to [tex]\( b^0 = 1 \)[/tex], which is consistent.
For negative values of [tex]\( x \)[/tex] (e.g., [tex]\( x = -1 \)[/tex]):
[tex]\[ y = 3 \][/tex]
If [tex]\( y = b^{-1} \)[/tex] when [tex]\( x = -1 \)[/tex], then [tex]\( y = \frac{1}{b} = 3 \)[/tex], thus [tex]\( b = \frac{1}{3} \)[/tex].
If [tex]\( b = \frac{1}{3} \)[/tex]:
For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = \left(\frac{1}{3}\right)^1 = \frac{1}{3} \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \][/tex]
For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = \left(\frac{1}{3}\right)^{-2} = 9 \][/tex]
For [tex]\( x = -3 \)[/tex]:
[tex]\[ y = \left(\frac{1}{3}\right)^{-3} = 27 \][/tex]
These values match exactly with those in Table 2.
Thus, Table 2 correctly represents an exponential function of the form [tex]\( y = b^x \)[/tex] where [tex]\( b = \frac{1}{3} \)[/tex] and [tex]\( 0 < b < 1 \)[/tex].
Therefore, the table that represents the exponential decay function [tex]\( y = b^x \)[/tex] with [tex]\( 0 < b < 1 \)[/tex] is:
Table 2
