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To determine which set of ordered pairs could be generated by an exponential function, we'll examine each set of pairs to see if they follow the properties of an exponential function. An exponential function can be generally expressed as [tex]\( y = ab^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.
Let's analyze each set:
1. Set 1: [tex]\((1,1), \left(2, \frac{1}{2}\right), \left(3, \frac{1}{3}\right), \left(4, \frac{1}{4}\right)\)[/tex]
This set does not fit the form [tex]\( y = ab^x \)[/tex] for a constant ratio [tex]\( b \)[/tex] because:
- [tex]\(\frac{\frac{1}{2}}{1} = \frac{1}{2}\)[/tex]
- [tex]\(\frac{\frac{1}{3}}{\frac{1}{2}} = \frac{2}{3}\)[/tex]
- [tex]\(\frac{\frac{1}{4}}{\frac{1}{3}} = \frac{3}{4}\)[/tex]
The ratio is not constant.
2. Set 2: [tex]\((1,1), \left(2, \frac{1}{4}\right), \left(3, \frac{1}{9}\right), \left(4, \frac{1}{16}\right)\)[/tex]
This set also does not fit the form [tex]\( y = ab^x \)[/tex] because:
- [tex]\(\frac{\frac{1}{4}}{1} = \frac{1}{4}\)[/tex]
- [tex]\(\frac{\frac{1}{9}}{\frac{1}{4}} = \frac{4}{9}\)[/tex]
- [tex]\(\frac{\frac{1}{16}}{\frac{1}{9}} = \frac{9}{16}\)[/tex]
The ratio is not constant.
3. 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]
This set fits the form [tex]\( y = ab^x \)[/tex]:
- [tex]\(\frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}\)[/tex]
- [tex]\(\frac{\frac{1}{8}}{\frac{1}{4}} = \frac{1}{2}\)[/tex]
- [tex]\(\frac{\frac{1}{16}}{\frac{1}{8}} = \frac{1}{2}\)[/tex]
All ratios are consistent with a common ratio [tex]\( b = \frac{1}{2} \)[/tex]. Thus, this can be written as [tex]\( y = \frac{1}{2} \left(\frac{1}{2}\right)^x = \left(\frac{1}{2}\right)^{x+1} \)[/tex].
Hence, Set 3 is generated by an exponential function.
4. 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]
This set does not fit the form [tex]\( y = ab^x \)[/tex]:
- [tex]\(\frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}\)[/tex]
- [tex]\(\frac{\frac{1}{6}}{\frac{1}{4}} = \frac{2}{3}\)[/tex]
- [tex]\(\frac{\frac{1}{8}}{\frac{1}{6}} = \frac{3}{4}\)[/tex]
The ratio is not constant.
Therefore, the set of ordered pairs that could be generated by an exponential function is:
[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]
Let's analyze each set:
1. Set 1: [tex]\((1,1), \left(2, \frac{1}{2}\right), \left(3, \frac{1}{3}\right), \left(4, \frac{1}{4}\right)\)[/tex]
This set does not fit the form [tex]\( y = ab^x \)[/tex] for a constant ratio [tex]\( b \)[/tex] because:
- [tex]\(\frac{\frac{1}{2}}{1} = \frac{1}{2}\)[/tex]
- [tex]\(\frac{\frac{1}{3}}{\frac{1}{2}} = \frac{2}{3}\)[/tex]
- [tex]\(\frac{\frac{1}{4}}{\frac{1}{3}} = \frac{3}{4}\)[/tex]
The ratio is not constant.
2. Set 2: [tex]\((1,1), \left(2, \frac{1}{4}\right), \left(3, \frac{1}{9}\right), \left(4, \frac{1}{16}\right)\)[/tex]
This set also does not fit the form [tex]\( y = ab^x \)[/tex] because:
- [tex]\(\frac{\frac{1}{4}}{1} = \frac{1}{4}\)[/tex]
- [tex]\(\frac{\frac{1}{9}}{\frac{1}{4}} = \frac{4}{9}\)[/tex]
- [tex]\(\frac{\frac{1}{16}}{\frac{1}{9}} = \frac{9}{16}\)[/tex]
The ratio is not constant.
3. 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]
This set fits the form [tex]\( y = ab^x \)[/tex]:
- [tex]\(\frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}\)[/tex]
- [tex]\(\frac{\frac{1}{8}}{\frac{1}{4}} = \frac{1}{2}\)[/tex]
- [tex]\(\frac{\frac{1}{16}}{\frac{1}{8}} = \frac{1}{2}\)[/tex]
All ratios are consistent with a common ratio [tex]\( b = \frac{1}{2} \)[/tex]. Thus, this can be written as [tex]\( y = \frac{1}{2} \left(\frac{1}{2}\right)^x = \left(\frac{1}{2}\right)^{x+1} \)[/tex].
Hence, Set 3 is generated by an exponential function.
4. 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]
This set does not fit the form [tex]\( y = ab^x \)[/tex]:
- [tex]\(\frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}\)[/tex]
- [tex]\(\frac{\frac{1}{6}}{\frac{1}{4}} = \frac{2}{3}\)[/tex]
- [tex]\(\frac{\frac{1}{8}}{\frac{1}{6}} = \frac{3}{4}\)[/tex]
The ratio is not constant.
Therefore, the set of ordered pairs that could be generated by an exponential function is:
[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]
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