IDNLearn.com: Where your questions meet expert advice and community insights. Ask anything and get well-informed, reliable answers from our knowledgeable community members.
Sagot :
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
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. IDNLearn.com is committed to providing the best answers. Thank you for visiting, and see you next time for more solutions.
