Question

compare the functions $f(x) = 2x^4$ and $g(x) = 4^x$ by completing parts (a) and (b). (a) fill in the table below. note that the table is already filled in for $x = 3$. (the aleks calculator can be used to make computations easier.) | $x$ | $f(x) = 2x^4$ | $g(x) = 4^x$ | |---|---|---| | 3 | 162 | 64 | | 4 | $square$ | $square$ | | 5 | $square$ | $square$ | | 6 | $square$ | $square$ | | 7 | $square$ | $square$ | (b) for $x \geq 6$, the table suggests that $f(x)$ is select greater than $g(x)$.

Explanation:

Part (a)

Step 1: Calculate \( f(x) = 2x^4 \) and \( g(x) = 4^x \) for \( x = 4 \)

• For \( f(4) \):
• Substitute \( x = 4 \) into \( f(x) = 2x^4 \).
• \( f(4) = 2\times(4)^4 = 2\times256 = 512 \)
• For \( g(4) \):
• Substitute \( x = 4 \) into \( g(x) = 4^x \).
• \( g(4) = 4^4 = 256 \)

Step 2: Calculate \( f(x) = 2x^4 \) and \( g(x) = 4^x \) for \( x = 5 \)

• For \( f(5) \):
• Substitute \( x = 5 \) into \( f(x) = 2x^4 \).
• \( f(5) = 2\times(5)^4 = 2\times625 = 1250 \)
• For \( g(5) \):
• Substitute \( x = 5 \) into \( g(x) = 4^x \).
• \( g(5) = 4^5 = 1024 \)

Step 3: Calculate \( f(x) = 2x^4 \) and \( g(x) = 4^x \) for \( x = 6 \)

• For \( f(6) \):
• Substitute \( x = 6 \) into \( f(x) = 2x^4 \).
• \( f(6) = 2\times(6)^4 = 2\times1296 = 2592 \)
• For \( g(6) \):
• Substitute \( x = 6 \) into \( g(x) = 4^x \).
• \( g(6) = 4^6 = 4096 \)

Step 4: Calculate \( f(x) = 2x^4 \) and \( g(x) = 4^x \) for \( x = 7 \)

• For \( f(7) \):
• Substitute \( x = 7 \) into \( f(x) = 2x^4 \).
• \( f(7) = 2\times(7)^4 = 2\times2401 = 4802 \)
• For \( g(7) \):
• Substitute \( x = 7 \) into \( g(x) = 4^x \).
• \( g(7) = 4^7 = 16384 \)

Now, we fill the table:

\( x \)	\( f(x) = 2x^4 \)	\( g(x) = 4^x \)
4	512	256
5	1250	1024
6	2592	4096
7	4802	16384

Part (b)

For \( x \geq 6 \), we compare \( f(x) \) and \( g(x) \):

• When \( x = 6 \): \( f(6) = 2592 \) and \( g(6) = 4096 \), so \( f(6) < g(6) \)
• When \( x = 7 \): \( f(7) = 4802 \) and \( g(7) = 16384 \), so \( f(7) < g(7) \)

So, for \( x \geq 6 \), the table suggests that \( f(x) \) is not greater than \( g(x) \).

Final Answers

Part (a) Table:

\( x \)	\( f(x) = 2x^4 \)	\( g(x) = 4^x \)
4	512	256
5	1250	1024
6	2592	4096
7	4802	16384

Part (b)

The table suggests that \( f(x) \) is \(\boldsymbol{\text{not}}\) greater than \( g(x) \) for \( x \geq 6 \).

Answer:

Part (a)

Step 1: Calculate \( f(x) = 2x^4 \) and \( g(x) = 4^x \) for \( x = 4 \)

• For \( f(4) \):
• Substitute \( x = 4 \) into \( f(x) = 2x^4 \).
• \( f(4) = 2\times(4)^4 = 2\times256 = 512 \)
• For \( g(4) \):
• Substitute \( x = 4 \) into \( g(x) = 4^x \).
• \( g(4) = 4^4 = 256 \)

Step 2: Calculate \( f(x) = 2x^4 \) and \( g(x) = 4^x \) for \( x = 5 \)

• For \( f(5) \):
• Substitute \( x = 5 \) into \( f(x) = 2x^4 \).
• \( f(5) = 2\times(5)^4 = 2\times625 = 1250 \)
• For \( g(5) \):
• Substitute \( x = 5 \) into \( g(x) = 4^x \).
• \( g(5) = 4^5 = 1024 \)

Step 3: Calculate \( f(x) = 2x^4 \) and \( g(x) = 4^x \) for \( x = 6 \)

• For \( f(6) \):
• Substitute \( x = 6 \) into \( f(x) = 2x^4 \).
• \( f(6) = 2\times(6)^4 = 2\times1296 = 2592 \)
• For \( g(6) \):
• Substitute \( x = 6 \) into \( g(x) = 4^x \).
• \( g(6) = 4^6 = 4096 \)

Step 4: Calculate \( f(x) = 2x^4 \) and \( g(x) = 4^x \) for \( x = 7 \)

• For \( f(7) \):
• Substitute \( x = 7 \) into \( f(x) = 2x^4 \).
• \( f(7) = 2\times(7)^4 = 2\times2401 = 4802 \)
• For \( g(7) \):
• Substitute \( x = 7 \) into \( g(x) = 4^x \).
• \( g(7) = 4^7 = 16384 \)

Now, we fill the table:

\( x \)	\( f(x) = 2x^4 \)	\( g(x) = 4^x \)
4	512	256
5	1250	1024
6	2592	4096
7	4802	16384

Part (b)

For \( x \geq 6 \), we compare \( f(x) \) and \( g(x) \):

• When \( x = 6 \): \( f(6) = 2592 \) and \( g(6) = 4096 \), so \( f(6) < g(6) \)
• When \( x = 7 \): \( f(7) = 4802 \) and \( g(7) = 16384 \), so \( f(7) < g(7) \)

So, for \( x \geq 6 \), the table suggests that \( f(x) \) is not greater than \( g(x) \).

Final Answers

Part (a) Table:

\( x \)	\( f(x) = 2x^4 \)	\( g(x) = 4^x \)
4	512	256
5	1250	1024
6	2592	4096
7	4802	16384

Part (b)

The table suggests that \( f(x) \) is \(\boldsymbol{\text{not}}\) greater than \( g(x) \) for \( x \geq 6 \).