Discover the answers you need at Westonci.ca, a dynamic Q&A platform where knowledge is shared freely by a community of experts. Discover detailed solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
