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Sagot :
To determine which of the given functions is an exponential function, we need to evaluate the nature of the sequences. An exponential function has the general form [tex]\( a \times b^x \)[/tex], meaning its ratios between consecutive terms should be consistent (i.e., the ratio [tex]\( \frac{f(x+1)}{f(x)} \)[/tex] should be constant for all [tex]\( x \)[/tex]).
Let's check each function in the table:
### Evaluate [tex]\( f(x) \)[/tex]
[tex]\[ \begin{array}{c|c|c|c|c|c|c|c} x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline f(x) & 2 & 5 & 10 & 17 & 26 & 37 & 50 \\ \end{array} \][/tex]
Calculate the ratios:
[tex]\[ \frac{5}{2} = 2.5, \quad \frac{10}{5} = 2, \quad \frac{17}{10} = 1.7, \quad \frac{26}{17} \approx 1.53, \quad \frac{37}{26} \approx 1.42, \quad \frac{50}{37} \approx 1.35 \][/tex]
The ratios are not consistent; hence, [tex]\( f(x) \)[/tex] is not an exponential function.
### Evaluate [tex]\( g(x) \)[/tex]
[tex]\[ \begin{array}{c|c|c|c|c|c|c|c} x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline g(x) & 0.125 & 0.25 & 0.5 & 1 & 2 & 4 & 8 \\ \end{array} \][/tex]
Calculate the ratios:
[tex]\[ \frac{0.25}{0.125} = 2, \quad \frac{0.5}{0.25} = 2, \quad \frac{1}{0.5} = 2, \quad \frac{2}{1} = 2, \quad \frac{4}{2} = 2, \quad \frac{8}{4} = 2 \][/tex]
The ratios are consistent; hence, [tex]\( g(x) \)[/tex] is an exponential function.
### Evaluate [tex]\( h(x) \)[/tex]
[tex]\[ \begin{array}{c|c|c|c|c|c|c|c} x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline h(x) & 0.25 & 0.5 & 0.75 & 1 & 1.25 & 1.5 & 1.75 \\ \end{array} \][/tex]
Calculate the ratios:
[tex]\[ \frac{0.5}{0.25} = 2, \quad \frac{0.75}{0.5} = 1.5, \quad \frac{1}{0.75} \approx 1.33, \quad \frac{1.25}{1} = 1.25, \quad \frac{1.5}{1.25} = 1.2, \quad \frac{1.75}{1.5} \approx 1.17 \][/tex]
The ratios are not consistent; hence, [tex]\( h(x) \)[/tex] is not an exponential function.
### Evaluate [tex]\( k(x) \)[/tex]
[tex]\[ \begin{array}{c|c|c|c|c|c|c|c} x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline k(x) & 64 & 16 & 4 & 1 & 0.25 & 0.0625 & 0.0156 \\ \end{array} \][/tex]
Calculate the ratios:
[tex]\[ \frac{16}{64} = 0.25, \quad \frac{4}{16} = 0.25, \quad \frac{1}{4} = 0.25, \quad \frac{0.25}{1} = 0.25, \quad \frac{0.0625}{0.25} = 0.25, \quad \frac{0.0156}{0.0625} \approx 0.25 \][/tex]
The ratios are consistent; however, [tex]\( k(x) \)[/tex] is formatted as a decreasing exponential function. Given the prompt's requirement for traditional exponential growth, this interpretation of exponential reduction might not apply directly.
From the ratios and consistent checking:
- [tex]\( f(x) \)[/tex] is not an exponential function.
- [tex]\( g(x) \)[/tex] is an exponential function.
- [tex]\( h(x) \)[/tex] is not an exponential function.
- [tex]\( k(x) \)[/tex] might indicate exponential decay but interpreting it isn't explicitly required in the way common exponential growth is understood.
Thus, the function that can confidently be labeled as an exponential function among the given is:
[tex]\[ \boxed{\text{A. } g(x)} \][/tex]
Let's check each function in the table:
### Evaluate [tex]\( f(x) \)[/tex]
[tex]\[ \begin{array}{c|c|c|c|c|c|c|c} x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline f(x) & 2 & 5 & 10 & 17 & 26 & 37 & 50 \\ \end{array} \][/tex]
Calculate the ratios:
[tex]\[ \frac{5}{2} = 2.5, \quad \frac{10}{5} = 2, \quad \frac{17}{10} = 1.7, \quad \frac{26}{17} \approx 1.53, \quad \frac{37}{26} \approx 1.42, \quad \frac{50}{37} \approx 1.35 \][/tex]
The ratios are not consistent; hence, [tex]\( f(x) \)[/tex] is not an exponential function.
### Evaluate [tex]\( g(x) \)[/tex]
[tex]\[ \begin{array}{c|c|c|c|c|c|c|c} x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline g(x) & 0.125 & 0.25 & 0.5 & 1 & 2 & 4 & 8 \\ \end{array} \][/tex]
Calculate the ratios:
[tex]\[ \frac{0.25}{0.125} = 2, \quad \frac{0.5}{0.25} = 2, \quad \frac{1}{0.5} = 2, \quad \frac{2}{1} = 2, \quad \frac{4}{2} = 2, \quad \frac{8}{4} = 2 \][/tex]
The ratios are consistent; hence, [tex]\( g(x) \)[/tex] is an exponential function.
### Evaluate [tex]\( h(x) \)[/tex]
[tex]\[ \begin{array}{c|c|c|c|c|c|c|c} x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline h(x) & 0.25 & 0.5 & 0.75 & 1 & 1.25 & 1.5 & 1.75 \\ \end{array} \][/tex]
Calculate the ratios:
[tex]\[ \frac{0.5}{0.25} = 2, \quad \frac{0.75}{0.5} = 1.5, \quad \frac{1}{0.75} \approx 1.33, \quad \frac{1.25}{1} = 1.25, \quad \frac{1.5}{1.25} = 1.2, \quad \frac{1.75}{1.5} \approx 1.17 \][/tex]
The ratios are not consistent; hence, [tex]\( h(x) \)[/tex] is not an exponential function.
### Evaluate [tex]\( k(x) \)[/tex]
[tex]\[ \begin{array}{c|c|c|c|c|c|c|c} x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline k(x) & 64 & 16 & 4 & 1 & 0.25 & 0.0625 & 0.0156 \\ \end{array} \][/tex]
Calculate the ratios:
[tex]\[ \frac{16}{64} = 0.25, \quad \frac{4}{16} = 0.25, \quad \frac{1}{4} = 0.25, \quad \frac{0.25}{1} = 0.25, \quad \frac{0.0625}{0.25} = 0.25, \quad \frac{0.0156}{0.0625} \approx 0.25 \][/tex]
The ratios are consistent; however, [tex]\( k(x) \)[/tex] is formatted as a decreasing exponential function. Given the prompt's requirement for traditional exponential growth, this interpretation of exponential reduction might not apply directly.
From the ratios and consistent checking:
- [tex]\( f(x) \)[/tex] is not an exponential function.
- [tex]\( g(x) \)[/tex] is an exponential function.
- [tex]\( h(x) \)[/tex] is not an exponential function.
- [tex]\( k(x) \)[/tex] might indicate exponential decay but interpreting it isn't explicitly required in the way common exponential growth is understood.
Thus, the function that can confidently be labeled as an exponential function among the given is:
[tex]\[ \boxed{\text{A. } g(x)} \][/tex]
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