IDNLearn.com is the perfect place to get answers, share knowledge, and learn new things. Ask any question and get a thorough, accurate answer from our community of experienced professionals.
Sagot :
To determine which table represents an exponential function, let's examine the nature of exponential functions and how they manifest through given data points.
An exponential function has the form:
[tex]\[ f(x) = a \cdot b^x \][/tex]
where [tex]\( b \)[/tex] is a constant base and [tex]\( a \)[/tex] is a constant coefficient.
For each table, we'll check if the ratio of consecutive [tex]\( f(x) \)[/tex]-values remains constant, a key characteristic of exponential growth or decay.
### Table 1 Analysis:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 5 \\ \hline 3 & 8 \\ \hline 4 & 11 \\ \hline \end{array} \][/tex]
Ratios,
[tex]\[ \frac{f(1)}{f(0)} = \frac{3}{1} = 3, \quad \frac{f(2)}{f(1)} = \frac{5}{3}, \quad \frac{f(3)}{f(2)} = \frac{8}{5}, \quad \frac{f(4)}{f(3)} = \frac{11}{8} \][/tex]
These ratios are not equal. Therefore, Table 1 does not represent an exponential function.
### Table 2 Analysis:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 0 & 1 \\ \hline 1 & 4 \\ \hline 2 & 16 \\ \hline 3 & 64 \\ \hline 4 & 256 \\ \hline \end{array} \][/tex]
Ratios,
[tex]\[ \frac{f(1)}{f(0)} = \frac{4}{1} = 4, \quad \frac{f(2)}{f(1)} = \frac{16}{4} = 4, \quad \frac{f(3)}{f(2)} = \frac{64}{16} = 4, \quad \frac{f(4)}{f(3)} = \frac{256}{64} = 4 \][/tex]
These ratios are all equal. Therefore, Table 2 represents an exponential function with a base [tex]\( b = 4 \)[/tex].
### Table 3 Analysis:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 0 & 2 \\ \hline 1 & 4 \\ \hline 2 & 6 \\ \hline 3 & 10 \\ \hline 4 & 12 \\ \hline \end{array} \][/tex]
Ratios,
[tex]\[ \frac{f(1)}{f(0)} = \frac{4}{2} = 2, \quad \frac{f(2)}{f(1)} = \frac{6}{4} = 1.5, \quad \frac{f(3)}{f(2)} = \frac{10}{6} \approx 1.67, \quad \frac{f(4)}{f(3)} = \frac{12}{10} = 1.2 \][/tex]
These ratios are not equal. Therefore, Table 3 does not represent an exponential function.
### Conclusion:
Upon inspecting the ratios, it is evident that Table 2 has a consistent ratio between consecutive [tex]\( f(x) \)[/tex] values. Thus, Table 2 represents an exponential function.
An exponential function has the form:
[tex]\[ f(x) = a \cdot b^x \][/tex]
where [tex]\( b \)[/tex] is a constant base and [tex]\( a \)[/tex] is a constant coefficient.
For each table, we'll check if the ratio of consecutive [tex]\( f(x) \)[/tex]-values remains constant, a key characteristic of exponential growth or decay.
### Table 1 Analysis:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 5 \\ \hline 3 & 8 \\ \hline 4 & 11 \\ \hline \end{array} \][/tex]
Ratios,
[tex]\[ \frac{f(1)}{f(0)} = \frac{3}{1} = 3, \quad \frac{f(2)}{f(1)} = \frac{5}{3}, \quad \frac{f(3)}{f(2)} = \frac{8}{5}, \quad \frac{f(4)}{f(3)} = \frac{11}{8} \][/tex]
These ratios are not equal. Therefore, Table 1 does not represent an exponential function.
### Table 2 Analysis:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 0 & 1 \\ \hline 1 & 4 \\ \hline 2 & 16 \\ \hline 3 & 64 \\ \hline 4 & 256 \\ \hline \end{array} \][/tex]
Ratios,
[tex]\[ \frac{f(1)}{f(0)} = \frac{4}{1} = 4, \quad \frac{f(2)}{f(1)} = \frac{16}{4} = 4, \quad \frac{f(3)}{f(2)} = \frac{64}{16} = 4, \quad \frac{f(4)}{f(3)} = \frac{256}{64} = 4 \][/tex]
These ratios are all equal. Therefore, Table 2 represents an exponential function with a base [tex]\( b = 4 \)[/tex].
### Table 3 Analysis:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 0 & 2 \\ \hline 1 & 4 \\ \hline 2 & 6 \\ \hline 3 & 10 \\ \hline 4 & 12 \\ \hline \end{array} \][/tex]
Ratios,
[tex]\[ \frac{f(1)}{f(0)} = \frac{4}{2} = 2, \quad \frac{f(2)}{f(1)} = \frac{6}{4} = 1.5, \quad \frac{f(3)}{f(2)} = \frac{10}{6} \approx 1.67, \quad \frac{f(4)}{f(3)} = \frac{12}{10} = 1.2 \][/tex]
These ratios are not equal. Therefore, Table 3 does not represent an exponential function.
### Conclusion:
Upon inspecting the ratios, it is evident that Table 2 has a consistent ratio between consecutive [tex]\( f(x) \)[/tex] values. Thus, Table 2 represents an exponential function.
Your participation is crucial to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. Your questions deserve precise answers. Thank you for visiting IDNLearn.com, and see you again soon for more helpful information.
