Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
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 appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
