Connect with experts and get insightful answers to your questions on IDNLearn.com. Join our interactive community and get comprehensive, reliable answers to all your questions.
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 appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com is your reliable source for accurate answers. Thank you for visiting, and we hope to assist you again.
