Answer :
To solve this problem, let's analyze and compare the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex].
### Function [tex]\( f \)[/tex]
We have the function [tex]\( f \)[/tex] given by a table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 15 & 7 & 3 & 1 & 0 \\ \hline \end{array} \][/tex]
We need to check whether [tex]\( f \)[/tex] is decreasing on the interval [tex]\([0,3]\)[/tex].
- For [tex]\( x \)[/tex] from 0 to 1: [tex]\( f(0) = 15 \)[/tex] and [tex]\( f(1) = 7 \)[/tex]. Since [tex]\( 15 > 7 \)[/tex], [tex]\( f \)[/tex] is decreasing on this interval.
- For [tex]\( x \)[/tex] from 1 to 2: [tex]\( f(1) = 7 \)[/tex] and [tex]\( f(2) = 3 \)[/tex]. Since [tex]\( 7 > 3 \)[/tex], [tex]\( f \)[/tex] is decreasing on this interval.
- For [tex]\( x \)[/tex] from 2 to 3: [tex]\( f(2) = 3 \)[/tex] and [tex]\( f(3) = 1 \)[/tex]. Since [tex]\( 3 > 1 \)[/tex], [tex]\( f \)[/tex] is decreasing on this interval.
Thus, [tex]\( f \)[/tex] is decreasing on [tex]\([0,3]\)[/tex].
Next, we check if [tex]\( f \)[/tex] is positive on [tex]\([0,3]\)[/tex]:
- For [tex]\( x = 0 \)[/tex] to [tex]\( x = 3 \)[/tex], [tex]\( f(x) \)[/tex] values are [tex]\( 15, 7, 3, 1 \)[/tex], all of which are positive.
Therefore, [tex]\( f \)[/tex] is both decreasing and positive on [tex]\([0,3]\)[/tex].
### Function [tex]\( g \)[/tex]
The function [tex]\( g \)[/tex] is an exponential function passing through the points [tex]\((0,9)\)[/tex] and [tex]\((3,0)\)[/tex].
An exponential function generally has the form [tex]\( g(x) = a \cdot b^x \)[/tex].
Given [tex]\( g(0) = 9 \)[/tex]:
[tex]\[ g(0) = 9 \implies a = 9 \][/tex]
The function passes through [tex]\((3, 0)\)[/tex], and we can infer that the function approaches 0 as [tex]\( x \)[/tex] increases. For practical purposes and for comparison, assume an extremely small base [tex]\( b \)[/tex] near 0.
Checking if [tex]\( g \)[/tex] is decreasing on [tex]\([0,3]\)[/tex]:
- Since the base [tex]\( b \)[/tex] is less than 1, [tex]\( g(x) \)[/tex] is decreasing as [tex]\( x \)[/tex] increases because each subsequent value of [tex]\( g(x) \)[/tex] becomes smaller.
### Rate of Decrease Comparison
To determine which function is decreasing at a faster rate, consider the actual changes between consecutive values for [tex]\( f \)[/tex] and [tex]\( g \)[/tex]:
For [tex]\( f \)[/tex]:
[tex]\[ \begin{align*} f(0) - f(1) &= 15 - 7 = 8, \\ f(1) - f(2) &= 7 - 3 = 4, \\ f(2) - f(3) &= 3 - 1 = 2. \end{align*} \][/tex]
For [tex]\( g \)[/tex], using a very small [tex]\( b \)[/tex], the difference between consecutive values will be:
[tex]\[ \begin{align*} g(0) - g(1) &= 9 - (9 \cdot b) = 9(1 - b), \\ g(1) - g(2) &= 9 \cdot b - 9 \cdot b^2 = 9b(1 - b), \\ g(2) - g(3) &= 9 \cdot b^2 - 9 \cdot b^3 = 9b^2(1 - b). \end{align*} \][/tex]
Given very small [tex]\( b \)[/tex], these values will be significantly smaller than the differences observed in [tex]\( f \)[/tex].
Thus, [tex]\( f \)[/tex] is decreasing at a faster rate than [tex]\( g \)[/tex].
### Conclusion
Based on our analysis:
- Both functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are decreasing on [tex]\([0,3]\)[/tex].
- Function [tex]\( f \)[/tex] is decreasing at a faster rate than [tex]\( g \)[/tex].
Therefore, the correct choice is:
[tex]\[ \boxed{C} \][/tex]
### Function [tex]\( f \)[/tex]
We have the function [tex]\( f \)[/tex] given by a table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 15 & 7 & 3 & 1 & 0 \\ \hline \end{array} \][/tex]
We need to check whether [tex]\( f \)[/tex] is decreasing on the interval [tex]\([0,3]\)[/tex].
- For [tex]\( x \)[/tex] from 0 to 1: [tex]\( f(0) = 15 \)[/tex] and [tex]\( f(1) = 7 \)[/tex]. Since [tex]\( 15 > 7 \)[/tex], [tex]\( f \)[/tex] is decreasing on this interval.
- For [tex]\( x \)[/tex] from 1 to 2: [tex]\( f(1) = 7 \)[/tex] and [tex]\( f(2) = 3 \)[/tex]. Since [tex]\( 7 > 3 \)[/tex], [tex]\( f \)[/tex] is decreasing on this interval.
- For [tex]\( x \)[/tex] from 2 to 3: [tex]\( f(2) = 3 \)[/tex] and [tex]\( f(3) = 1 \)[/tex]. Since [tex]\( 3 > 1 \)[/tex], [tex]\( f \)[/tex] is decreasing on this interval.
Thus, [tex]\( f \)[/tex] is decreasing on [tex]\([0,3]\)[/tex].
Next, we check if [tex]\( f \)[/tex] is positive on [tex]\([0,3]\)[/tex]:
- For [tex]\( x = 0 \)[/tex] to [tex]\( x = 3 \)[/tex], [tex]\( f(x) \)[/tex] values are [tex]\( 15, 7, 3, 1 \)[/tex], all of which are positive.
Therefore, [tex]\( f \)[/tex] is both decreasing and positive on [tex]\([0,3]\)[/tex].
### Function [tex]\( g \)[/tex]
The function [tex]\( g \)[/tex] is an exponential function passing through the points [tex]\((0,9)\)[/tex] and [tex]\((3,0)\)[/tex].
An exponential function generally has the form [tex]\( g(x) = a \cdot b^x \)[/tex].
Given [tex]\( g(0) = 9 \)[/tex]:
[tex]\[ g(0) = 9 \implies a = 9 \][/tex]
The function passes through [tex]\((3, 0)\)[/tex], and we can infer that the function approaches 0 as [tex]\( x \)[/tex] increases. For practical purposes and for comparison, assume an extremely small base [tex]\( b \)[/tex] near 0.
Checking if [tex]\( g \)[/tex] is decreasing on [tex]\([0,3]\)[/tex]:
- Since the base [tex]\( b \)[/tex] is less than 1, [tex]\( g(x) \)[/tex] is decreasing as [tex]\( x \)[/tex] increases because each subsequent value of [tex]\( g(x) \)[/tex] becomes smaller.
### Rate of Decrease Comparison
To determine which function is decreasing at a faster rate, consider the actual changes between consecutive values for [tex]\( f \)[/tex] and [tex]\( g \)[/tex]:
For [tex]\( f \)[/tex]:
[tex]\[ \begin{align*} f(0) - f(1) &= 15 - 7 = 8, \\ f(1) - f(2) &= 7 - 3 = 4, \\ f(2) - f(3) &= 3 - 1 = 2. \end{align*} \][/tex]
For [tex]\( g \)[/tex], using a very small [tex]\( b \)[/tex], the difference between consecutive values will be:
[tex]\[ \begin{align*} g(0) - g(1) &= 9 - (9 \cdot b) = 9(1 - b), \\ g(1) - g(2) &= 9 \cdot b - 9 \cdot b^2 = 9b(1 - b), \\ g(2) - g(3) &= 9 \cdot b^2 - 9 \cdot b^3 = 9b^2(1 - b). \end{align*} \][/tex]
Given very small [tex]\( b \)[/tex], these values will be significantly smaller than the differences observed in [tex]\( f \)[/tex].
Thus, [tex]\( f \)[/tex] is decreasing at a faster rate than [tex]\( g \)[/tex].
### Conclusion
Based on our analysis:
- Both functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are decreasing on [tex]\([0,3]\)[/tex].
- Function [tex]\( f \)[/tex] is decreasing at a faster rate than [tex]\( g \)[/tex].
Therefore, the correct choice is:
[tex]\[ \boxed{C} \][/tex]
