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Sagot :
To determine which set of ordered pairs can be generated by an exponential function, we need to verify if the ratio of consecutive y-values remains consistent across the selected pairs.
### Set 1: [tex]\((1,1), \left(2, \frac{1}{2}\right), \left(3, \frac{1}{3}\right), \left(4, \frac{1}{4}\right)\)[/tex]
- Ratios between consecutive y-values:
[tex]\[ \frac{\frac{1}{2}}{1} = \frac{1}{2}, \quad \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{2}{3}, \quad \frac{\frac{1}{4}}{\frac{1}{3}} = \frac{3}{4} \][/tex]
Since the ratios are different ([tex]\(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}\)[/tex]), this set of pairs cannot be generated by an exponential function.
### Set 2: [tex]\((1,1), \left(2, \frac{1}{4}\right), \left(3, \frac{1}{9}\right), \left(4, \frac{1}{16}\right)\)[/tex]
- Ratios between consecutive y-values:
[tex]\[ \frac{\frac{1}{4}}{1} = \frac{1}{4}, \quad \frac{\frac{1}{9}}{\frac{1}{4}} = \frac{4}{9}, \quad \frac{\frac{1}{16}}{\frac{1}{9}} = \frac{9}{16} \][/tex]
Since the ratios are different ([tex]\(\frac{1}{4}, \frac{4}{9}, \frac{9}{16}\)[/tex]), this set of pairs cannot be generated by an exponential function.
### Set 3: [tex]\(\left(1, \frac{1}{2}\right), \left(2, \frac{1}{4}\right), \left(3, \frac{1}{8}\right), \left(4, \frac{1}{16}\right)\)[/tex]
- Ratios between consecutive y-values:
[tex]\[ \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}, \quad \frac{\frac{1}{8}}{\frac{1}{4}} = \frac{1}{2}, \quad \frac{\frac{1}{16}}{\frac{1}{8}} = \frac{1}{2} \][/tex]
Since the ratios are all the same ([tex]\(\frac{1}{2}\)[/tex]), this set of pairs could indeed be generated by an exponential function.
### Set 4: [tex]\(\left(1, \frac{1}{2}\right), \left(2, \frac{1}{4}\right), \left(3, \frac{1}{6}\right), \left(4, \frac{1}{8}\right)\)[/tex]
- Ratios between consecutive y-values:
[tex]\[ \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}, \quad \frac{\frac{1}{6}}{\frac{1}{4}} = \frac{2}{3}, \quad \frac{\frac{1}{8}}{\frac{1}{6}} = \frac{3}{4} \][/tex]
Since the ratios are different ([tex]\(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}\)[/tex]), this set of pairs cannot be generated by an exponential function.
### Conclusion
Among the listed sets, only Set 3: [tex]\(\left(1, \frac{1}{2}\right), \left(2, \frac{1}{4}\right), \left(3, \frac{1}{8}\right), \left(4, \frac{1}{16}\right)\)[/tex] could be generated by an exponential function due to the consistent ratio between consecutive y-values.
### Set 1: [tex]\((1,1), \left(2, \frac{1}{2}\right), \left(3, \frac{1}{3}\right), \left(4, \frac{1}{4}\right)\)[/tex]
- Ratios between consecutive y-values:
[tex]\[ \frac{\frac{1}{2}}{1} = \frac{1}{2}, \quad \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{2}{3}, \quad \frac{\frac{1}{4}}{\frac{1}{3}} = \frac{3}{4} \][/tex]
Since the ratios are different ([tex]\(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}\)[/tex]), this set of pairs cannot be generated by an exponential function.
### Set 2: [tex]\((1,1), \left(2, \frac{1}{4}\right), \left(3, \frac{1}{9}\right), \left(4, \frac{1}{16}\right)\)[/tex]
- Ratios between consecutive y-values:
[tex]\[ \frac{\frac{1}{4}}{1} = \frac{1}{4}, \quad \frac{\frac{1}{9}}{\frac{1}{4}} = \frac{4}{9}, \quad \frac{\frac{1}{16}}{\frac{1}{9}} = \frac{9}{16} \][/tex]
Since the ratios are different ([tex]\(\frac{1}{4}, \frac{4}{9}, \frac{9}{16}\)[/tex]), this set of pairs cannot be generated by an exponential function.
### Set 3: [tex]\(\left(1, \frac{1}{2}\right), \left(2, \frac{1}{4}\right), \left(3, \frac{1}{8}\right), \left(4, \frac{1}{16}\right)\)[/tex]
- Ratios between consecutive y-values:
[tex]\[ \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}, \quad \frac{\frac{1}{8}}{\frac{1}{4}} = \frac{1}{2}, \quad \frac{\frac{1}{16}}{\frac{1}{8}} = \frac{1}{2} \][/tex]
Since the ratios are all the same ([tex]\(\frac{1}{2}\)[/tex]), this set of pairs could indeed be generated by an exponential function.
### Set 4: [tex]\(\left(1, \frac{1}{2}\right), \left(2, \frac{1}{4}\right), \left(3, \frac{1}{6}\right), \left(4, \frac{1}{8}\right)\)[/tex]
- Ratios between consecutive y-values:
[tex]\[ \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}, \quad \frac{\frac{1}{6}}{\frac{1}{4}} = \frac{2}{3}, \quad \frac{\frac{1}{8}}{\frac{1}{6}} = \frac{3}{4} \][/tex]
Since the ratios are different ([tex]\(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}\)[/tex]), this set of pairs cannot be generated by an exponential function.
### Conclusion
Among the listed sets, only Set 3: [tex]\(\left(1, \frac{1}{2}\right), \left(2, \frac{1}{4}\right), \left(3, \frac{1}{8}\right), \left(4, \frac{1}{16}\right)\)[/tex] could be generated by an exponential function due to the consistent ratio between consecutive y-values.
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