Answer :
### Question 1: What type of function is [tex]\( B(x) \)[/tex], linear, quadratic, or exponential? Justify your answer and show calculations to support your conclusion.
To determine the type of function for [tex]\( B(x) \)[/tex], we analyze the given values for [tex]\( B(x) \)[/tex] at different years:
- [tex]\( B(0) \)[/tex]
- [tex]\( B(1) = 6 \)[/tex]
- [tex]\( B(2) = 18 \)[/tex]
- [tex]\( B(3) = 54 \)[/tex]
- [tex]\( B(5) = 162 \)[/tex]
#### Step-by-Step Justification:
1. Calculate the Ratios:
- Ratio of [tex]\( B(1) \)[/tex] to [tex]\( B(0) \)[/tex] is undefined since [tex]\( B(0) \)[/tex] is not provided.
- Ratio of [tex]\( B(2) \)[/tex] to [tex]\( B(1) \)[/tex]:
[tex]\[ \frac{B(2)}{B(1)} = \frac{18}{6} = 3 \][/tex]
- Ratio of [tex]\( B(3) \)[/tex] to [tex]\( B(2) \)[/tex]:
[tex]\[ \frac{B(3)}{B(2)} = \frac{54}{18} = 3 \][/tex]
- Ratio of [tex]\( B(5) \)[/tex] to [tex]\( B(3) \)[/tex]:
[tex]\[ \frac{B(5)}{B(3)} = \frac{162}{54} = 3 \][/tex]
2. Conclusion:
The ratios [tex]\( \frac{B(2)}{B(1)} = 3 \)[/tex], [tex]\( \frac{B(3)}{B(2)} = 3 \)[/tex], and [tex]\( \frac{B(5)}{B(3)} = 3 \)[/tex] are constant. This constant ratio indicates that [tex]\( B(x) \)[/tex] is an exponential function.
Therefore, [tex]\( B(x) \)[/tex] is an exponential function with a common ratio of 3.
### Question 2: What type of function is [tex]\( M(x) \)[/tex], linear, quadratic, or exponential? Justify your answer and show calculations to support your conclusion.
To determine the type of function for [tex]\( M(x) \)[/tex], we analyze the given values for [tex]\( M(x) \)[/tex] at different years:
- [tex]\( M(0) \)[/tex]
- [tex]\( M(1) = 20 \)[/tex]
- [tex]\( M(2) = 120 \)[/tex]
- [tex]\( M(3) = 420 \)[/tex]
- [tex]\( M(5) = 920 \)[/tex]
#### Step-by-Step Justification:
1. Calculate the First Differences:
- Difference between [tex]\( M(2) \)[/tex] and [tex]\( M(1) \)[/tex]:
[tex]\[ M(2) - M(1) = 120 - 20 = 100 \][/tex]
- Difference between [tex]\( M(3) \)[/tex] and [tex]\( M(2) \)[/tex]:
[tex]\[ M(3) - M(2) = 420 - 120 = 300 \][/tex]
- Difference between [tex]\( M(5) \)[/tex] and [tex]\( M(3) \)[/tex]:
[tex]\[ M(5) - M(3) = 920 - 420 = 500 \][/tex]
2. Analyze the Differences:
The first differences [tex]\( 100 \)[/tex], [tex]\( 300 \)[/tex], and [tex]\( 500 \)[/tex] are not constant. This indicates that [tex]\( M(x) \)[/tex] is not a linear function.
3. Calculate the Ratios:
- Ratio of [tex]\( M(2) \)[/tex] to [tex]\( M(1) \)[/tex]:
[tex]\[ \frac{M(2)}{M(1)} = \frac{120}{20} = 6 \][/tex]
- Ratio of [tex]\( M(3) \)[/tex] to [tex]\( M(2) \)[/tex]:
[tex]\[ \frac{M(3)}{M(2)} = \frac{420}{120} = 3.5 \][/tex]
- Ratio of [tex]\( M(5) \)[/tex] to [tex]\( M(3) \)[/tex]:
[tex]\[ \frac{M(5)}{M(3)} = \frac{920}{420} \approx 2.19 \][/tex]
4. Conclusion:
The ratios [tex]\( \frac{M(2)}{M(1)} = 6 \)[/tex], [tex]\( \frac{M(3)}{M(2)} = 3.5 \)[/tex], and [tex]\( \frac{M(5)}{M(3)} \approx 2.19 \)[/tex] are not constant. This indicates that [tex]\( M(x) \)[/tex] is neither exponential nor linear. [tex]\( M(x) \)[/tex] is likely a higher-order polynomial function or another type of non-linear function.
Therefore, [tex]\( M(x) \)[/tex] is neither linear nor exponential and appears to be a non-linear function based on its variable differences and ratios.
To determine the type of function for [tex]\( B(x) \)[/tex], we analyze the given values for [tex]\( B(x) \)[/tex] at different years:
- [tex]\( B(0) \)[/tex]
- [tex]\( B(1) = 6 \)[/tex]
- [tex]\( B(2) = 18 \)[/tex]
- [tex]\( B(3) = 54 \)[/tex]
- [tex]\( B(5) = 162 \)[/tex]
#### Step-by-Step Justification:
1. Calculate the Ratios:
- Ratio of [tex]\( B(1) \)[/tex] to [tex]\( B(0) \)[/tex] is undefined since [tex]\( B(0) \)[/tex] is not provided.
- Ratio of [tex]\( B(2) \)[/tex] to [tex]\( B(1) \)[/tex]:
[tex]\[ \frac{B(2)}{B(1)} = \frac{18}{6} = 3 \][/tex]
- Ratio of [tex]\( B(3) \)[/tex] to [tex]\( B(2) \)[/tex]:
[tex]\[ \frac{B(3)}{B(2)} = \frac{54}{18} = 3 \][/tex]
- Ratio of [tex]\( B(5) \)[/tex] to [tex]\( B(3) \)[/tex]:
[tex]\[ \frac{B(5)}{B(3)} = \frac{162}{54} = 3 \][/tex]
2. Conclusion:
The ratios [tex]\( \frac{B(2)}{B(1)} = 3 \)[/tex], [tex]\( \frac{B(3)}{B(2)} = 3 \)[/tex], and [tex]\( \frac{B(5)}{B(3)} = 3 \)[/tex] are constant. This constant ratio indicates that [tex]\( B(x) \)[/tex] is an exponential function.
Therefore, [tex]\( B(x) \)[/tex] is an exponential function with a common ratio of 3.
### Question 2: What type of function is [tex]\( M(x) \)[/tex], linear, quadratic, or exponential? Justify your answer and show calculations to support your conclusion.
To determine the type of function for [tex]\( M(x) \)[/tex], we analyze the given values for [tex]\( M(x) \)[/tex] at different years:
- [tex]\( M(0) \)[/tex]
- [tex]\( M(1) = 20 \)[/tex]
- [tex]\( M(2) = 120 \)[/tex]
- [tex]\( M(3) = 420 \)[/tex]
- [tex]\( M(5) = 920 \)[/tex]
#### Step-by-Step Justification:
1. Calculate the First Differences:
- Difference between [tex]\( M(2) \)[/tex] and [tex]\( M(1) \)[/tex]:
[tex]\[ M(2) - M(1) = 120 - 20 = 100 \][/tex]
- Difference between [tex]\( M(3) \)[/tex] and [tex]\( M(2) \)[/tex]:
[tex]\[ M(3) - M(2) = 420 - 120 = 300 \][/tex]
- Difference between [tex]\( M(5) \)[/tex] and [tex]\( M(3) \)[/tex]:
[tex]\[ M(5) - M(3) = 920 - 420 = 500 \][/tex]
2. Analyze the Differences:
The first differences [tex]\( 100 \)[/tex], [tex]\( 300 \)[/tex], and [tex]\( 500 \)[/tex] are not constant. This indicates that [tex]\( M(x) \)[/tex] is not a linear function.
3. Calculate the Ratios:
- Ratio of [tex]\( M(2) \)[/tex] to [tex]\( M(1) \)[/tex]:
[tex]\[ \frac{M(2)}{M(1)} = \frac{120}{20} = 6 \][/tex]
- Ratio of [tex]\( M(3) \)[/tex] to [tex]\( M(2) \)[/tex]:
[tex]\[ \frac{M(3)}{M(2)} = \frac{420}{120} = 3.5 \][/tex]
- Ratio of [tex]\( M(5) \)[/tex] to [tex]\( M(3) \)[/tex]:
[tex]\[ \frac{M(5)}{M(3)} = \frac{920}{420} \approx 2.19 \][/tex]
4. Conclusion:
The ratios [tex]\( \frac{M(2)}{M(1)} = 6 \)[/tex], [tex]\( \frac{M(3)}{M(2)} = 3.5 \)[/tex], and [tex]\( \frac{M(5)}{M(3)} \approx 2.19 \)[/tex] are not constant. This indicates that [tex]\( M(x) \)[/tex] is neither exponential nor linear. [tex]\( M(x) \)[/tex] is likely a higher-order polynomial function or another type of non-linear function.
Therefore, [tex]\( M(x) \)[/tex] is neither linear nor exponential and appears to be a non-linear function based on its variable differences and ratios.
