Question

compare the functions $f(x)=2^x$ and $g(x)=4x^3$ by completing parts (a) and (b).
(a) fill in the table below. note that the table is already filled in for $x = 5$.
(the aleks calculator can be used to make computations easier.)

$x$	$f(x)=2^x$	$g(x)=4x^3$
6	$square$	$square$
13	$square$	$square$
14	$square$	$square$
15	$square$	$square$

(b) for $x \geq 14$, the table suggests that $f(x)$ is select greater than $g(x)$.

Explanation:

Step1: Calculate \( f(x) = 2^x \) and \( g(x) = 4x^3 \) for \( x = 6 \)

For \( f(6) \): \( f(6)=2^6 = 64 \)
For \( g(6) \): \( g(6)=4\times6^3=4\times216 = 864 \)

Step2: Calculate \( f(x) = 2^x \) and \( g(x) = 4x^3 \) for \( x = 13 \)

For \( f(13) \): \( f(13)=2^{13}=8192 \)
For \( g(13) \): \( g(13)=4\times13^3=4\times2197 = 8788 \)

Step3: Calculate \( f(x) = 2^x \) and \( g(x) = 4x^3 \) for \( x = 14 \)

For \( f(14) \): \( f(14)=2^{14}=16384 \)
For \( g(14) \): \( g(14)=4\times14^3=4\times2744 = 10976 \)

Step4: Calculate \( f(x) = 2^x \) and \( g(x) = 4x^3 \) for \( x = 15 \)

For \( f(15) \): \( f(15)=2^{15}=32768 \)
For \( g(15) \): \( g(15)=4\times15^3=4\times3375 = 13500 \)

Step5: Analyze part (b)

Compare \( f(x) \) and \( g(x) \) for \( x\geq14 \). From step 3 and 4:

• At \( x = 14 \), \( f(14)=16384 \), \( g(14)=10976 \), so \( f(14)>g(14) \)
• At \( x = 15 \), \( f(15)=32768 \), \( g(15)=13500 \), so \( f(15)>g(15) \)

So for \( x\geq14 \), \( f(x) \) is greater than \( g(x) \)

Answer:

(a)

\( x \)	\( f(x)=2^x \)	\( g(x)=4x^3 \)
13	8192	8788
14	16384	10976
15	32768	13500

(b) For \( x\geq14 \), the table suggests that \( f(x) \) is \(\boldsymbol{\text{always}}\) greater than \( g(x) \) (the "Select" dropdown should be filled with "always" or similar indicating \( f(x) > g(x) \) for \( x\geq14 \))