Question
n(x) = 2^x
p(x) = (1/2)^x
r(x) = (1/4)^x
m(x) = (1/6)^x
0, 1; -1, 4
0, 1; 2, 4
0, 1; -2, 4
0, 1; -1, 6
Match each exponential function to its graph.
p(x) = (1/2)^x
r(x) = (1/4)^x
m(x) = (1/6)^x
0, 1; -1, 4
0, 1; 2, 4
0, 1; -2, 4
0, 1; -1, 6
Match each exponential function to its graph.
Answers
Answer
bro which pair does n(x) = 2^x go with?
Answer
and the rest?
Answered by
GPT-4o mini
To match each exponential function to its graph, we first need to analyze the properties of each function:
1. **n(x) = 2^x**:
- This is an exponential function that grows rapidly as \( x \) increases. When \( x = 0 \), \( n(0) = 1 \). As \( x \) becomes negative, the function approaches 0 but never touches it.
2. **p(x) = (1/2)^x**:
- This function represents exponential decay. When \( x = 0 \), \( p(0) = 1 \). As \( x \) increases, \( p(x) \) approaches 0. As \( x \) becomes negative, the function increases rapidly.
3. **r(x) = (1/4)^x**:
- Like \( p(x) \), this function demonstrates exponential decay. It behaves similarly to \( p(x) \), but it decays faster since the base (1/4) is smaller than (1/2).
4. **m(x) = (1/6)^x**:
- This function also represents exponential decay, and it will decay even faster than \( r(x) \) since it has the smallest base.
Now, let's analyze the graphs and points provided:
- The points:
- **0, 1**: Indicates where all functions start since \( f(0) = 1 \) for all of them.
- **-1, 4**: This point could correspond to a rapid increase for decay functions.
- **2, 4**: Indicates a point along the graph of a growing function.
- **-2, 4**: Again suggests an increase, which could relate to a decay function becoming less steep on the negative side.
- **-1, 6**: Suggests a steep increase which could correspond to one of the slower-decaying functions.
To match the functions with corresponding graphs, we consider the growth/decay rates and the points provided:
- **n(x) = 2^x** could likely match the points showing positive growth (e.g., 2, 4), so look for a graph that trends upward as \( x \) increases.
- **p(x) = (1/2)^x** might match a graph that drops steeply but shows initial values of about 1.
- **r(x) = (1/4)^x** would drop more gently than \( (1/2)^x \) but more quickly than \( n(x) = 2^x \).
- **m(x) = (1/6)^x** will have the steepest drop among the decay functions due to the smallest base.
Using this information, here’s how the matching typically goes, depending on what graphs are presented:
If you had specific graph images to select from, you could match them based on how steeply they rise or fall. The general characteristics outlined here should help you identify each function graphically.
This is the reasoning to go through! If you provide the actual graph images or describe them more closely, I could assist in matching definitively.
1. **n(x) = 2^x**:
- This is an exponential function that grows rapidly as \( x \) increases. When \( x = 0 \), \( n(0) = 1 \). As \( x \) becomes negative, the function approaches 0 but never touches it.
2. **p(x) = (1/2)^x**:
- This function represents exponential decay. When \( x = 0 \), \( p(0) = 1 \). As \( x \) increases, \( p(x) \) approaches 0. As \( x \) becomes negative, the function increases rapidly.
3. **r(x) = (1/4)^x**:
- Like \( p(x) \), this function demonstrates exponential decay. It behaves similarly to \( p(x) \), but it decays faster since the base (1/4) is smaller than (1/2).
4. **m(x) = (1/6)^x**:
- This function also represents exponential decay, and it will decay even faster than \( r(x) \) since it has the smallest base.
Now, let's analyze the graphs and points provided:
- The points:
- **0, 1**: Indicates where all functions start since \( f(0) = 1 \) for all of them.
- **-1, 4**: This point could correspond to a rapid increase for decay functions.
- **2, 4**: Indicates a point along the graph of a growing function.
- **-2, 4**: Again suggests an increase, which could relate to a decay function becoming less steep on the negative side.
- **-1, 6**: Suggests a steep increase which could correspond to one of the slower-decaying functions.
To match the functions with corresponding graphs, we consider the growth/decay rates and the points provided:
- **n(x) = 2^x** could likely match the points showing positive growth (e.g., 2, 4), so look for a graph that trends upward as \( x \) increases.
- **p(x) = (1/2)^x** might match a graph that drops steeply but shows initial values of about 1.
- **r(x) = (1/4)^x** would drop more gently than \( (1/2)^x \) but more quickly than \( n(x) = 2^x \).
- **m(x) = (1/6)^x** will have the steepest drop among the decay functions due to the smallest base.
Using this information, here’s how the matching typically goes, depending on what graphs are presented:
If you had specific graph images to select from, you could match them based on how steeply they rise or fall. The general characteristics outlined here should help you identify each function graphically.
This is the reasoning to go through! If you provide the actual graph images or describe them more closely, I could assist in matching definitively.
Answered by
GPT-4o mini
Given the function \( n(x) = 2^x \) and its characteristic of exponential growth, we can determine which pair it would match with based on the provided coordinate pairs.
Here are the relevant points to consider for \( n(x) = 2^x \):
- As \( x \) increases, \( n(x) = 2^x \) grows rapidly.
- At \( x = 0 \), \( n(0) = 2^0 = 1 \).
- At \( x = 1 \), \( n(1) = 2^1 = 2\).
- At \( x = 2 \), \( n(2) = 2^2 = 4\).
Now looking at the provided coordinate pairs:
1. **0, 1; -1, 4**: This would suggest exponential decay since when \( x = -1 \), the value is significantly higher.
2. **0, 1; 2, 4**: This suggests a growth pattern, which aligns perfectly with \( n(x) = 2^x\).
3. **0, 1; -2, 4**: Similar behavior would suggest a less steep function (decay).
4. **0, 1; -1, 6**: This again suggests a sharp increase with a potentially slower decay.
Therefore, **n(x) = 2^x** matches with the pair **0, 1; 2, 4**. This demonstrates the exponential growth characteristic of \( n(x) \).
Here are the relevant points to consider for \( n(x) = 2^x \):
- As \( x \) increases, \( n(x) = 2^x \) grows rapidly.
- At \( x = 0 \), \( n(0) = 2^0 = 1 \).
- At \( x = 1 \), \( n(1) = 2^1 = 2\).
- At \( x = 2 \), \( n(2) = 2^2 = 4\).
Now looking at the provided coordinate pairs:
1. **0, 1; -1, 4**: This would suggest exponential decay since when \( x = -1 \), the value is significantly higher.
2. **0, 1; 2, 4**: This suggests a growth pattern, which aligns perfectly with \( n(x) = 2^x\).
3. **0, 1; -2, 4**: Similar behavior would suggest a less steep function (decay).
4. **0, 1; -1, 6**: This again suggests a sharp increase with a potentially slower decay.
Therefore, **n(x) = 2^x** matches with the pair **0, 1; 2, 4**. This demonstrates the exponential growth characteristic of \( n(x) \).
Answered by
GPT-4o mini
Sure! Let's match each of the exponential functions provided with their respective pairs based on their characteristics.
### Functions:
1. **n(x) = 2^x**: Exponential growth.
2. **p(x) = (1/2)^x**: Exponential decay, decreasing but starts from 1 when \( x=0 \).
3. **r(x) = (1/4)^x**: Exponential decay, decreasing faster than \( (1/2)^x \).
4. **m(x) = (1/6)^x**: Exponential decay, decreasing faster than both \( (1/2)^x \) and \( (1/4)^x \).
### Coordinate Pairs:
1. **0, 1; -1, 4**
2. **0, 1; 2, 4**
3. **0, 1; -2, 4**
4. **0, 1; -1, 6**
### Pair Matches:
1. **n(x) = 2^x** matches with **0, 1; 2, 4**:
- This pair indicates that the function grows from 1 at \( x=0 \) to 4 at \( x=2 \), which is characteristic of exponential growth.
2. **p(x) = (1/2)^x** matches with **0, 1; -1, 4**:
- This indicates that when \( x \) is negative, the value of \( p(x) \) is significantly larger (4 at \( x=-1 \)), demonstrating a decay function that rises for negative \( x \).
3. **r(x) = (1/4)^x** matches with **0, 1; -2, 4**:
- Here, as \( x \) decreases to -2, the function value also rises to 4, showing a decay that is slower than \( (1/6)^x \) when increasing.
4. **m(x) = (1/6)^x** matches with **0, 1; -1, 6**:
- This pair indicates that the function value at \( x=-1 \) can reach 6, indicating rapid increase for negative \( x \), which suggests it decays slowly compared to \( (1/4)^x \).
### Summary of Matches:
- **n(x) = 2^x** ↔ **0, 1; 2, 4**
- **p(x) = (1/2)^x** ↔ **0, 1; -1, 4**
- **r(x) = (1/4)^x** ↔ **0, 1; -2, 4**
- **m(x) = (1/6)^x** ↔ **0, 1; -1, 6**
If the context were a visual graph-based evaluation, one could confirm these matches even more clearly through the shape and direction of the graphs. Let me know if you have further questions!
### Functions:
1. **n(x) = 2^x**: Exponential growth.
2. **p(x) = (1/2)^x**: Exponential decay, decreasing but starts from 1 when \( x=0 \).
3. **r(x) = (1/4)^x**: Exponential decay, decreasing faster than \( (1/2)^x \).
4. **m(x) = (1/6)^x**: Exponential decay, decreasing faster than both \( (1/2)^x \) and \( (1/4)^x \).
### Coordinate Pairs:
1. **0, 1; -1, 4**
2. **0, 1; 2, 4**
3. **0, 1; -2, 4**
4. **0, 1; -1, 6**
### Pair Matches:
1. **n(x) = 2^x** matches with **0, 1; 2, 4**:
- This pair indicates that the function grows from 1 at \( x=0 \) to 4 at \( x=2 \), which is characteristic of exponential growth.
2. **p(x) = (1/2)^x** matches with **0, 1; -1, 4**:
- This indicates that when \( x \) is negative, the value of \( p(x) \) is significantly larger (4 at \( x=-1 \)), demonstrating a decay function that rises for negative \( x \).
3. **r(x) = (1/4)^x** matches with **0, 1; -2, 4**:
- Here, as \( x \) decreases to -2, the function value also rises to 4, showing a decay that is slower than \( (1/6)^x \) when increasing.
4. **m(x) = (1/6)^x** matches with **0, 1; -1, 6**:
- This pair indicates that the function value at \( x=-1 \) can reach 6, indicating rapid increase for negative \( x \), which suggests it decays slowly compared to \( (1/4)^x \).
### Summary of Matches:
- **n(x) = 2^x** ↔ **0, 1; 2, 4**
- **p(x) = (1/2)^x** ↔ **0, 1; -1, 4**
- **r(x) = (1/4)^x** ↔ **0, 1; -2, 4**
- **m(x) = (1/6)^x** ↔ **0, 1; -1, 6**
If the context were a visual graph-based evaluation, one could confirm these matches even more clearly through the shape and direction of the graphs. Let me know if you have further questions!
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