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 rate of \( g(x) = 3 \times 4^x \) with \( f(x) \), we need to examine the functions more closely.
1. **Growth Rate of \( g(x) \)**:
The function \( g(x) = 3 \times 4^x \) grows exponentially due to the \( 4^x \) term. The growth rate is primarily determined by the base of the exponent, which is \( 4 \).
2. **Assuming \( f(x) \) is a function of the same form**:
If we assume \( f(x) \) is also an exponential function of the form \( f(x) = k \times a^x \) for some constants \( k \) and \( a \), the growth rate comparison would depend on the bases \( 4 \) and \( a \).
3. **Comparing Growth Rates**:
If we wanted to express the growth rate of \( g(x) \) in terms of \( f(x) \), notice that the derivative of an exponential function is proportional to itself multiplied by the natural logarithm of the base. The growth rate (or the rate of change) is significant for comparing \( g(x) \) with exponential \( f(x) \).
Assuming \( f(x) = k \times a^x \):
The exponential growth rates are:
- \( g'(x) \) has a growth rate of \( 3 \times 4^x \times \ln(4) \)
- \( f'(x) \) has a growth rate of \( k \times a^x \times \ln(a) \)
To find how \( g \) grows compared to \( f \), compute:
\[
\frac{g'(x)}{f'(x)} = \frac{3 \times 4^x \times \ln(4)}{k \times a^x \times \ln(a)}
\]
The ratio involves \( \frac{4^x}{a^x} \), which simplifies to \( (4/a)^x \).
### Conclusion:
Given \( g(x) \) has a base of \( 4 \), the growth is exponentially faster than any base less than \( 4 \).
From the choices provided, without specific knowledge of \( f(x) \), but with the understanding that \( 4^x \) has a constant multiplier of \( 3 \), we note that the growth rate of \( g(x) \) could relate to \( 6^x \) based on the options provided in terms of base comparisons.
After careful comparison, the best representation of the relationship given multiple choices is:
**The growth rate of \( g(x) \) is \( 2^x \) times the growth rate of \( f(x) \), emphasizing how much faster it grows due to its base component.**
Therefore, the correct answer is:
**D. It is \( 2^x \) times the growth rate of \( f(x) \).**
1. **Growth Rate of \( g(x) \)**:
The function \( g(x) = 3 \times 4^x \) grows exponentially due to the \( 4^x \) term. The growth rate is primarily determined by the base of the exponent, which is \( 4 \).
2. **Assuming \( f(x) \) is a function of the same form**:
If we assume \( f(x) \) is also an exponential function of the form \( f(x) = k \times a^x \) for some constants \( k \) and \( a \), the growth rate comparison would depend on the bases \( 4 \) and \( a \).
3. **Comparing Growth Rates**:
If we wanted to express the growth rate of \( g(x) \) in terms of \( f(x) \), notice that the derivative of an exponential function is proportional to itself multiplied by the natural logarithm of the base. The growth rate (or the rate of change) is significant for comparing \( g(x) \) with exponential \( f(x) \).
Assuming \( f(x) = k \times a^x \):
The exponential growth rates are:
- \( g'(x) \) has a growth rate of \( 3 \times 4^x \times \ln(4) \)
- \( f'(x) \) has a growth rate of \( k \times a^x \times \ln(a) \)
To find how \( g \) grows compared to \( f \), compute:
\[
\frac{g'(x)}{f'(x)} = \frac{3 \times 4^x \times \ln(4)}{k \times a^x \times \ln(a)}
\]
The ratio involves \( \frac{4^x}{a^x} \), which simplifies to \( (4/a)^x \).
### Conclusion:
Given \( g(x) \) has a base of \( 4 \), the growth is exponentially faster than any base less than \( 4 \).
From the choices provided, without specific knowledge of \( f(x) \), but with the understanding that \( 4^x \) has a constant multiplier of \( 3 \), we note that the growth rate of \( g(x) \) could relate to \( 6^x \) based on the options provided in terms of base comparisons.
After careful comparison, the best representation of the relationship given multiple choices is:
**The growth rate of \( g(x) \) is \( 2^x \) times the growth rate of \( f(x) \), emphasizing how much faster it grows due to its base component.**
Therefore, the correct answer is:
**D. It is \( 2^x \) times the growth rate of \( f(x) \).**