Let's analyze and compare the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] step by step.
### Function Definitions
- [tex]\( f(x) = 40(1.3)^x \)[/tex]
- [tex]\( g(x) \)[/tex] is provided via a table:
[tex]\[
\begin{array}{|c|c|}
\hline x & g(x) \\
\hline 0 & 40 \\
\hline 1 & 56 \\
\hline 2 & 78.4 \\
\hline 3 & 109.76 \\
\hline
\end{array}
\][/tex]
### Initial Values
Let's compare the initial values (i.e., when [tex]\( x = 0 \)[/tex]):
For [tex]\( f(x) \)[/tex]:
[tex]\[
f(0) = 40 \cdot (1.3)^0 = 40 \cdot 1 = 40
\][/tex]
For [tex]\( g(x) \)[/tex], the table shows [tex]\( g(0) = 40 \)[/tex].
The initial values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are the same. Therefore, statement B is correct.
### Growth/Decay Factors
#### For [tex]\( f(x) \)[/tex]:
The growth factor in [tex]\( f(x) \)[/tex] is explicitly given by the base of the exponent, [tex]\( 1.3 \)[/tex]. Hence the growth factor is [tex]\( 1.3 \)[/tex].
#### For [tex]\( g(x) \)[/tex]:
We need to calculate the growth factor by examining the ratios of successive values.
[tex]\[
\frac{g(1)}{g(0)} = \frac{56}{40} = 1.4
\][/tex]
[tex]\[
\frac{g(2)}{g(1)} = \frac{78.4}{56} = 1.4
\][/tex]
[tex]\[
\frac{g(3)}{g(2)} = \frac{109.76}{78.4} = 1.4
\][/tex]
We observe that the growth factor for [tex]\( g(x) \)[/tex] is consistent and is [tex]\( 1.4 \)[/tex].
### Comparisons
Comparing the growth factors:
- The growth factor of [tex]\( f(x) \)[/tex] is [tex]\( 1.3 \)[/tex].
- The growth factor of [tex]\( g(x) \)[/tex] is [tex]\( 1.4 \)[/tex].
Clearly, [tex]\( 1.3 < 1.4 \)[/tex], so the growth factor of [tex]\( f(x) \)[/tex] is less than that of [tex]\( g(x) \)[/tex]. Therefore, statement C is correct.
### Decay Factors
Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are growing functions (since their growth factors are greater than 1). Therefore, statements involving decay factors are not applicable here. Hence, statement A is not correct.
### Final Verification:
- Statement B: Correct. The initial values are 40 for both functions.
- Statement C: Correct. The growth factor of [tex]\( f(x) \)[/tex] is less than that of [tex]\( g(x) \)[/tex].
- Statement D: Incorrect. The growth factor of [tex]\( f(x) \)[/tex] is not more than that of [tex]\( g(x) \)[/tex].
### Answer
Based on our analysis, the correct statements are:
[tex]\[
B. \text{The initial values of } f(x) \text{ and } g(x) \text{ are the same.}
\][/tex]
[tex]\[
C. \text{The growth factor of } f(x) \text{ is less than the growth factor of } g(x).
\][/tex]
