Question
g of x equals 3 times 4 Superscript x Baseline
How does the growth rate of g of x compare to the growth rate of f of x at any given x value?
Answer options with 5 options
A.
It is 2 times the growth rate of f of x.
B.
It is 6 times the growth rate of f of x.
C.
It is 16 times the growth rate of f of x.
D.
It is 2 Superscript x Baseline times the growth rate of f of x.
E.
It is 6 Superscript x Baseline times the growth rate of f of x.
How does the growth rate of g of x compare to the growth rate of f of x at any given x value?
Answer options with 5 options
A.
It is 2 times the growth rate of f of x.
B.
It is 6 times the growth rate of f of x.
C.
It is 16 times the growth rate of f of x.
D.
It is 2 Superscript x Baseline times the growth rate of f of x.
E.
It is 6 Superscript x Baseline times the growth rate of f of x.
Answers
GPT-4o mini
To compare the growth rates of \( g(x) = 3 \cdot 4^x \) with the function \( f(x) \), we first need to clarify what function \( f(x) \) represents.
However, since \( f(x) \) is not specified in your question, let's assume that \( f(x) \) is a simple exponential function of the form \( f(x) = a \cdot b^x \), where \( a \) and \( b \) are constants, with \( b \) being the base of the exponential growth.
If we set \( f(x) = a \cdot b^x \) as a general exponential growth function, we can calculate the growth rates of both functions.
The growth rate of an exponential function is generally proportional to the function itself. The key aspect of growth rates is determined by the derivative with respect to \( x \).
1. The derivative \( g'(x) \) for \( g(x) \):
\[
g(x) = 3 \cdot 4^x
\]
Using the chain rule,
\[
g'(x) = 3 \cdot (\ln(4)) \cdot 4^x
\]
2. The derivative \( f'(x) \) for \( f(x) \):
\[
f(x) = a \cdot b^x
\]
Similarly,
\[
f'(x) = a \cdot (\ln(b)) \cdot b^x
\]
To compare the growth rates:
\[
\frac{g'(x)}{f'(x)} = \frac{3 \cdot \ln(4) \cdot 4^x}{a \cdot \ln(b) \cdot b^x}
\]
For the growth rates between \( g(x) \) and \( f(x) \) to be comparable, you can set \( b \) as a value that can help simplify the comparison. If \( b \) is 4 (which matches the base of our \( g(x) \)), then we could see more direct relationships. However, without specific values for \( a \) and \( b \) being present in the problem, we can only derive a relative ratio that may not yield a clear numerical multiple.
If \( f(x) = 1 \cdot 4^x \), the ratio simplifies to:
\[
\frac{g'(x)}{f'(x)} = \frac{3 \ln(4)}{\ln(4)} = 3
\]
This indicates that the growth rate is 3 times that of \( f(x) \) at least when comparing to that specific chosen form of \( f(x) \).
If we want to explore the closest fits to the given answer options:
- Without knowing \( a \) and \( b \) we must generalize, but the best-fit option is likely to involve an interpretation of either base growth or constant multiplicative growth, based on \( a \) in \( f(x) \).
Given this reasoning and without the specific function \( f(x) \), if a constant compares:
- Therefore, we could conclude generally that \( g(x) \) expresses a constant factor, upon evaluation.
None of the supplied answer choices exactly fit without knowing \( f(x) \). Based on general growth analysis:
**Closer inspection might yield more light on specific match but from what you have, choose approximately as 6 times or exponential impact based off derived forms.**
Deriving specifics would allow to yield true comparisons, thus reputable citation for analysis is required from function \( f(x) \).
However, since \( f(x) \) is not specified in your question, let's assume that \( f(x) \) is a simple exponential function of the form \( f(x) = a \cdot b^x \), where \( a \) and \( b \) are constants, with \( b \) being the base of the exponential growth.
If we set \( f(x) = a \cdot b^x \) as a general exponential growth function, we can calculate the growth rates of both functions.
The growth rate of an exponential function is generally proportional to the function itself. The key aspect of growth rates is determined by the derivative with respect to \( x \).
1. The derivative \( g'(x) \) for \( g(x) \):
\[
g(x) = 3 \cdot 4^x
\]
Using the chain rule,
\[
g'(x) = 3 \cdot (\ln(4)) \cdot 4^x
\]
2. The derivative \( f'(x) \) for \( f(x) \):
\[
f(x) = a \cdot b^x
\]
Similarly,
\[
f'(x) = a \cdot (\ln(b)) \cdot b^x
\]
To compare the growth rates:
\[
\frac{g'(x)}{f'(x)} = \frac{3 \cdot \ln(4) \cdot 4^x}{a \cdot \ln(b) \cdot b^x}
\]
For the growth rates between \( g(x) \) and \( f(x) \) to be comparable, you can set \( b \) as a value that can help simplify the comparison. If \( b \) is 4 (which matches the base of our \( g(x) \)), then we could see more direct relationships. However, without specific values for \( a \) and \( b \) being present in the problem, we can only derive a relative ratio that may not yield a clear numerical multiple.
If \( f(x) = 1 \cdot 4^x \), the ratio simplifies to:
\[
\frac{g'(x)}{f'(x)} = \frac{3 \ln(4)}{\ln(4)} = 3
\]
This indicates that the growth rate is 3 times that of \( f(x) \) at least when comparing to that specific chosen form of \( f(x) \).
If we want to explore the closest fits to the given answer options:
- Without knowing \( a \) and \( b \) we must generalize, but the best-fit option is likely to involve an interpretation of either base growth or constant multiplicative growth, based on \( a \) in \( f(x) \).
Given this reasoning and without the specific function \( f(x) \), if a constant compares:
- Therefore, we could conclude generally that \( g(x) \) expresses a constant factor, upon evaluation.
None of the supplied answer choices exactly fit without knowing \( f(x) \). Based on general growth analysis:
**Closer inspection might yield more light on specific match but from what you have, choose approximately as 6 times or exponential impact based off derived forms.**
Deriving specifics would allow to yield true comparisons, thus reputable citation for analysis is required from function \( f(x) \).