Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To determine which set of ordered pairs could be generated by an exponential function, we'll analyze each set and see if they fit the form of an exponential function, [tex]\( y = a \cdot b^x \)[/tex].
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]
Let's check if this set can be represented by an exponential function:
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 1 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = \frac{1}{2} \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( y = \frac{1}{3} \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( y = \frac{1}{4} \)[/tex]
Notice that the pattern in the [tex]\( y \)[/tex]-values is not exponential (it looks more like a harmonic sequence), so this set does not match an exponential function.
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]
Let's check this set for an exponential pattern:
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 1 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = \frac{1}{4} = \left(\frac{1}{2}\right)^2 \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( y = \frac{1}{9} = \left(\frac{1}{3}\right)^2 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( y = \frac{1}{16} = \left(\frac{1}{4}\right)^2 \)[/tex]
We can see that [tex]\( y = \left(\frac{1}{x}\right)^2 \)[/tex], which implies that [tex]\( y = x^{-2} \)[/tex], matching the form of an exponential function [tex]\( y = a \cdot b^x \)[/tex] with [tex]\( a = 1 \)[/tex] and [tex]\( b = x^{-2} \)[/tex].
3. Set 3: [tex]\(\left(1, \frac{1}{2}\right), \left(2, \frac{1}{4}\right), \left(2, \frac{1}{8}\right), \left(4, \frac{1}{16}\right)\)[/tex]
This set has a repeated [tex]\( x \)[/tex]-value:
- For [tex]\( x = 1 \)[/tex], [tex]\( y = \frac{1}{2} \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = \frac{1}{4} \)[/tex] and another [tex]\( y = \frac{1}{8} \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( y = \frac{1}{16} \)[/tex]
The presence of repeated and conflicting [tex]\(y \)[/tex]-values for [tex]\( x = 2 \)[/tex] makes this inconsistent. Therefore, this set cannot represent 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]
Analyzing the [tex]\( y \)[/tex]-values for an exponential function:
- For [tex]\( x = 1 \)[/tex], [tex]\( y = \frac{1}{2} \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = \frac{1}{4} \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( y = \frac{1}{6} \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( y = \frac{1}{8} \)[/tex]
There's no clear exponential pattern here: for consistent exponential behavior, we'd expect a constant ratio between consecutive [tex]\( y \)[/tex]-values, which is not the case here ([tex]\( \frac{1/4}{1/2} \neq \frac{1/6}{1/4} \)[/tex]).
From this analysis, the set of ordered pairs that fits the pattern of an exponential function is:
Set 2: [tex]\((1,1), \left(2, \frac{1}{4}\right), \left(3, \frac{1}{9}\right), \left(4, \frac{1}{16}\right)\)[/tex]
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]
Let's check if this set can be represented by an exponential function:
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 1 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = \frac{1}{2} \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( y = \frac{1}{3} \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( y = \frac{1}{4} \)[/tex]
Notice that the pattern in the [tex]\( y \)[/tex]-values is not exponential (it looks more like a harmonic sequence), so this set does not match an exponential function.
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]
Let's check this set for an exponential pattern:
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 1 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = \frac{1}{4} = \left(\frac{1}{2}\right)^2 \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( y = \frac{1}{9} = \left(\frac{1}{3}\right)^2 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( y = \frac{1}{16} = \left(\frac{1}{4}\right)^2 \)[/tex]
We can see that [tex]\( y = \left(\frac{1}{x}\right)^2 \)[/tex], which implies that [tex]\( y = x^{-2} \)[/tex], matching the form of an exponential function [tex]\( y = a \cdot b^x \)[/tex] with [tex]\( a = 1 \)[/tex] and [tex]\( b = x^{-2} \)[/tex].
3. Set 3: [tex]\(\left(1, \frac{1}{2}\right), \left(2, \frac{1}{4}\right), \left(2, \frac{1}{8}\right), \left(4, \frac{1}{16}\right)\)[/tex]
This set has a repeated [tex]\( x \)[/tex]-value:
- For [tex]\( x = 1 \)[/tex], [tex]\( y = \frac{1}{2} \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = \frac{1}{4} \)[/tex] and another [tex]\( y = \frac{1}{8} \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( y = \frac{1}{16} \)[/tex]
The presence of repeated and conflicting [tex]\(y \)[/tex]-values for [tex]\( x = 2 \)[/tex] makes this inconsistent. Therefore, this set cannot represent 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]
Analyzing the [tex]\( y \)[/tex]-values for an exponential function:
- For [tex]\( x = 1 \)[/tex], [tex]\( y = \frac{1}{2} \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = \frac{1}{4} \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( y = \frac{1}{6} \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( y = \frac{1}{8} \)[/tex]
There's no clear exponential pattern here: for consistent exponential behavior, we'd expect a constant ratio between consecutive [tex]\( y \)[/tex]-values, which is not the case here ([tex]\( \frac{1/4}{1/2} \neq \frac{1/6}{1/4} \)[/tex]).
From this analysis, the set of ordered pairs that fits the pattern of an exponential function is:
Set 2: [tex]\((1,1), \left(2, \frac{1}{4}\right), \left(3, \frac{1}{9}\right), \left(4, \frac{1}{16}\right)\)[/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
