Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To solve this problem, we'll proceed with the following steps:
### Part (a)
We need to fill in the table by computing the values of the functions [tex]\( f(x) = 50x^2 \)[/tex] and [tex]\( g(x) = 4^x \)[/tex] for each given [tex]\( x \)[/tex].
Given in the table:
[tex]\[ \begin{array}{|c|c|c|} \hline x & f(x) = 50 x^2 & g(x) = 4^x \\ \hline 3 & 450 & 64 \\ \hline \end{array} \][/tex]
Let's compute the values for [tex]\( x =4, 5, 6, 7 \)[/tex]:
1. For [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = 50 \cdot 4^2 = 50 \cdot 16 = 800 \][/tex]
[tex]\[ g(4) = 4^4 = 256 \][/tex]
2. For [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = 50 \cdot 5^2 = 50 \cdot 25 = 1250 \][/tex]
[tex]\[ g(5) = 4^5 = 1024 \][/tex]
3. For [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = 50 \cdot 6^2 = 50 \cdot 36 = 1800 \][/tex]
[tex]\[ g(6) = 4^6 = 4096 \][/tex]
4. For [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = 50 \cdot 7^2 = 50 \cdot 49 = 2450 \][/tex]
[tex]\[ g(7) = 4^7 = 16384 \][/tex]
Now we fill in the table:
[tex]\[ \begin{tabular}{|c|c|c|} \hline x & f(x)=50 x^2 & g(x)=4^x \\ \hline 3 & 450 & 64 \\ \hline 4 & 800 & 256 \\ \hline 5 & 1250 & 1024 \\ \hline 6 & 1800 & 4096 \\ \hline 7 & 2450 & 16384 \\ \hline \end{tabular} \][/tex]
### Part (b)
We need to look at the values for [tex]\( x \geq 4 \)[/tex] and determine for which [tex]\( x \)[/tex] the function [tex]\( f(x) \)[/tex] is greater than [tex]\( g(x) \)[/tex].
From the table, for [tex]\( x \geq 4 \)[/tex]:
- For [tex]\( x = 4 \)[/tex], [tex]\( f(4) = 800 \)[/tex] and [tex]\( g(4) = 256 \)[/tex]. Here, [tex]\( f(4) > g(4) \)[/tex].
- For [tex]\( x = 5 \)[/tex], [tex]\( f(5) = 1250 \)[/tex] and [tex]\( g(5) = 1024 \)[/tex]. Here, [tex]\( f(5) > g(5) \)[/tex].
- For [tex]\( x = 6 \)[/tex], [tex]\( f(6) = 1800 \)[/tex] and [tex]\( g(6) = 4096 \)[/tex]. Here, [tex]\( f(6) < g(6) \)[/tex].
- For [tex]\( x = 7 \)[/tex], [tex]\( f(7) = 2450 \)[/tex] and [tex]\( g(7) = 16384 \)[/tex]. Here, [tex]\( f(7) < g(7) \)[/tex].
Therefore, for [tex]\( x \geq 4 \)[/tex], we observe that [tex]\( f(x) \)[/tex] is greater than [tex]\( g(x) \)[/tex] for [tex]\( x = 4 \)[/tex] and [tex]\( x = 5 \)[/tex], but [tex]\( f(x) \)[/tex] is less than [tex]\( g(x) \)[/tex] for [tex]\( x = 6 \)[/tex] and [tex]\( x = 7 \)[/tex].
Thus, the results for [tex]\( x \geq 4 \)[/tex] suggest that:
[tex]\[ \text{For } x \geq 4, \text{ the table suggests that } f(x) \text{ is greater than } g(x) \text{ for the values } x < 6. \][/tex]
The final statement can be rephrased as:
For [tex]\( x \geq 4 \)[/tex], the table suggests that [tex]\( f(x) \)[/tex] is:
[tex]\[ \boxed{\text{greater than } g(x) \text{ for some values and less for others.}} \][/tex]
### Part (a)
We need to fill in the table by computing the values of the functions [tex]\( f(x) = 50x^2 \)[/tex] and [tex]\( g(x) = 4^x \)[/tex] for each given [tex]\( x \)[/tex].
Given in the table:
[tex]\[ \begin{array}{|c|c|c|} \hline x & f(x) = 50 x^2 & g(x) = 4^x \\ \hline 3 & 450 & 64 \\ \hline \end{array} \][/tex]
Let's compute the values for [tex]\( x =4, 5, 6, 7 \)[/tex]:
1. For [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = 50 \cdot 4^2 = 50 \cdot 16 = 800 \][/tex]
[tex]\[ g(4) = 4^4 = 256 \][/tex]
2. For [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = 50 \cdot 5^2 = 50 \cdot 25 = 1250 \][/tex]
[tex]\[ g(5) = 4^5 = 1024 \][/tex]
3. For [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = 50 \cdot 6^2 = 50 \cdot 36 = 1800 \][/tex]
[tex]\[ g(6) = 4^6 = 4096 \][/tex]
4. For [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = 50 \cdot 7^2 = 50 \cdot 49 = 2450 \][/tex]
[tex]\[ g(7) = 4^7 = 16384 \][/tex]
Now we fill in the table:
[tex]\[ \begin{tabular}{|c|c|c|} \hline x & f(x)=50 x^2 & g(x)=4^x \\ \hline 3 & 450 & 64 \\ \hline 4 & 800 & 256 \\ \hline 5 & 1250 & 1024 \\ \hline 6 & 1800 & 4096 \\ \hline 7 & 2450 & 16384 \\ \hline \end{tabular} \][/tex]
### Part (b)
We need to look at the values for [tex]\( x \geq 4 \)[/tex] and determine for which [tex]\( x \)[/tex] the function [tex]\( f(x) \)[/tex] is greater than [tex]\( g(x) \)[/tex].
From the table, for [tex]\( x \geq 4 \)[/tex]:
- For [tex]\( x = 4 \)[/tex], [tex]\( f(4) = 800 \)[/tex] and [tex]\( g(4) = 256 \)[/tex]. Here, [tex]\( f(4) > g(4) \)[/tex].
- For [tex]\( x = 5 \)[/tex], [tex]\( f(5) = 1250 \)[/tex] and [tex]\( g(5) = 1024 \)[/tex]. Here, [tex]\( f(5) > g(5) \)[/tex].
- For [tex]\( x = 6 \)[/tex], [tex]\( f(6) = 1800 \)[/tex] and [tex]\( g(6) = 4096 \)[/tex]. Here, [tex]\( f(6) < g(6) \)[/tex].
- For [tex]\( x = 7 \)[/tex], [tex]\( f(7) = 2450 \)[/tex] and [tex]\( g(7) = 16384 \)[/tex]. Here, [tex]\( f(7) < g(7) \)[/tex].
Therefore, for [tex]\( x \geq 4 \)[/tex], we observe that [tex]\( f(x) \)[/tex] is greater than [tex]\( g(x) \)[/tex] for [tex]\( x = 4 \)[/tex] and [tex]\( x = 5 \)[/tex], but [tex]\( f(x) \)[/tex] is less than [tex]\( g(x) \)[/tex] for [tex]\( x = 6 \)[/tex] and [tex]\( x = 7 \)[/tex].
Thus, the results for [tex]\( x \geq 4 \)[/tex] suggest that:
[tex]\[ \text{For } x \geq 4, \text{ the table suggests that } f(x) \text{ is greater than } g(x) \text{ for the values } x < 6. \][/tex]
The final statement can be rephrased as:
For [tex]\( x \geq 4 \)[/tex], the table suggests that [tex]\( f(x) \)[/tex] is:
[tex]\[ \boxed{\text{greater than } g(x) \text{ for some values and less for others.}} \][/tex]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.