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Which of the following statements accurately compares [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex]? Choose all that apply.

[tex]\[ f(x) = 40(1.3)^x \][/tex]

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$g(x)$[/tex] \\
\hline
0 & 40 \\
\hline
1 & 56 \\
\hline
2 & 78.4 \\
\hline
3 & 109.76 \\
\hline
\end{tabular}

A. The decay factor of [tex]$f(x)$[/tex] is less than the decay factor of [tex]$g(x)$[/tex].

B. The initial values of [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex] are the same.

C. The growth factor of [tex]$f(x)$[/tex] is less than the growth factor of [tex]$g(x)$[/tex].

D. The growth factor of [tex]$f(x)$[/tex] is more than the growth factor of [tex]$g(x)$[/tex].

Sagot :

GinnyAnswer
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]
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