Question

compare the functions $f(x) = 3^x$ and $g(x) = 400x$ by completing parts (a) and (b).
(a) fill in the table below. note that the table is already filled in for $x = 4$.
(the aleks calculator can be used to make computations easier.)

x	$f(x) = 3^x$	$g(x) = 400x$
5
7
8
9

(b) for $x \geq 5$, the table suggests that $f(x)$ is select greater than $g(x)$.

Explanation:

Step1: Calculate \( f(5) \) and \( g(5) \)

For \( f(x) = 3^x \), when \( x = 5 \), \( f(5)=3^5 = 243 \). For \( g(x)=400x \), when \( x = 5 \), \( g(5)=400\times5 = 2000 \).

Step2: Calculate \( f(7) \) and \( g(7) \)

For \( f(x) = 3^x \), when \( x = 7 \), \( f(7)=3^7=2187 \). For \( g(x)=400x \), when \( x = 7 \), \( g(7)=400\times7 = 2800 \).

Step3: Calculate \( f(8) \) and \( g(8) \)

For \( f(x) = 3^x \), when \( x = 8 \), \( f(8)=3^8 = 6561 \). For \( g(x)=400x \), when \( x = 8 \), \( g(8)=400\times8 = 3200 \).

Step4: Calculate \( f(9) \) and \( g(9) \)

For \( f(x) = 3^x \), when \( x = 9 \), \( f(9)=3^9=19683 \). For \( g(x)=400x \), when \( x = 9 \), \( g(9)=400\times9 = 3600 \).

Step5: Analyze \( f(x) \) and \( g(x) \) for \( x\geq5 \)

Compare the values: At \( x = 5 \), \( f(5)=243 \), \( g(5)=2000 \) ( \( g \) is bigger). At \( x = 7 \), \( f(7)=2187 \), \( g(7)=2800 \) ( \( g \) is bigger). At \( x = 8 \), \( f(8)=6561 \), \( g(8)=3200 \) ( \( f \) is bigger). At \( x = 9 \), \( f(9)=19683 \), \( g(9)=3600 \) ( \( f \) is bigger). So after a certain point ( \( x\geq8 \) here, but the table shows as \( x\geq5 \) trend), but looking at \( x = 8,9 \), \( f(x) \) becomes greater. Wait, maybe the question is about when does \( f(x) \) become greater. But the table filling:

Filling the table:

• \( x = 5 \): \( f(5)=3^5 = 243 \), \( g(5)=400\times5 = 2000 \)
• \( x = 7 \): \( f(7)=3^7 = 2187 \), \( g(7)=400\times7 = 2800 \)
• \( x = 8 \): \( f(8)=3^8 = 6561 \), \( g(8)=400\times8 = 3200 \)
• \( x = 9 \): \( f(9)=3^9 = 19683 \), \( g(9)=400\times9 = 3600 \)

Then for part (b), when \( x\geq5 \), at \( x = 5,7 \), \( g(x) \) is bigger, at \( x = 8,9 \), \( f(x) \) is bigger. But maybe the question is as \( x \) increases, when does \( f(x) \) overtake \( g(x) \). From the table, at \( x = 8 \), \( f(8)=6561>3200 = g(8) \), at \( x = 9 \), \( f(9)>g(9) \). But the table's part (b) says "the table suggests that \( f(x) \) is [Select] greater than \( g(x) \)". Wait, maybe I miscalculated \( x = 7 \). Wait \( 3^7 = 2187 \), \( 400\times7 = 2800 \), so \( g(7) \) is bigger. \( x = 8 \): \( 3^8 = 6561 \), \( 400\times8 = 3200 \), so \( f(8) \) is bigger. So for \( x\geq8 \), but the table has \( x = 5,7,8,9 \). Wait the table is filled as:

x | \( f(x)=3^x \) | \( g(x)=400x \)
--- | --- | ---
4 | 81 | 1600
5 | 243 | 2000
7 | 2187 | 2800
8 | 6561 | 3200
9 | 19683 | 3600

Now, comparing \( f(x) \) and \( g(x) \):

• \( x = 5 \): 243 < 2000 (g bigger)
• \( x = 7 \): 2187 < 2800 (g bigger)
• \( x = 8 \): 6561 > 3200 (f bigger)
• \( x = 9 \): 19683 > 3600 (f bigger)

So the table suggests that for \( x\geq8 \), but the question says \( x\geq5 \). Wait maybe a typo, but the key is to fill the table.

Filled Table:

x	\( f(x)=3^x \)	\( g(x)=400x \)
5	243	2000
7	2187	2800
8	6561	3200
9	19683	3600

For part (b), when \( x\geq8 \) (from the table), \( f(x) \) is greater than \( g(x) \). But the question's dropdown is "Select" with options (probably "eventually" or "always" but from the table, at \( x = 5,7 \) it's not, at \( x = 8,9 \) it is. So the table suggests that as \( x \) increases (for \( x\geq8 \)), \( f(x) \) becomes greater. But the initial question's part (b) is "For \( x\geq5 \), the table suggests that \( f(x) \) is [Select] greater than \( g(x) \)". Wait maybe the user expects the filled table and the analysis.

Answer:

(for part a table filling):
For \( x = 5 \): \( f(5)=243 \), \( g(5)=2000 \)
For \( x = 7 \): \( f(7)=2187 \), \( g(7)=2800 \)
For \( x = 8 \): \( f(8)=6561 \), \( g(8)=3200 \)
For \( x = 9 \): \( f(9)=19683 \), \( g(9)=3600 \)

For part (b), the table suggests that for \( x\geq8 \) (or as \( x \) increases beyond 7), \( f(x) \) is greater than \( g(x) \). If the dropdown has "eventually" or "sometimes", but from the table, at \( x = 8,9 \), \( f(x) > g(x) \), so the answer for the dropdown (if options are "eventually" or "always") would be "eventually" (since it's not always, but becomes greater after a point).