Question

compare the functions $f(x) = 50x^2$ 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.)

$x$	$f(x) = 50x^2$	$g(x) = 4^x$
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 less than $g(x)$.

Explanation:

Step1: Calculate \( f(4) \) and \( g(4) \)

For \( f(x) = 50x^2 \), substitute \( x = 4 \): \( f(4)=50\times4^2 = 50\times16 = 800 \)
For \( g(x) = 4^x \), substitute \( x = 4 \): \( g(4)=4^4 = 256 \)

Step2: Calculate \( f(5) \) and \( g(5) \)

For \( f(x) = 50x^2 \), substitute \( x = 5 \): \( f(5)=50\times5^2 = 50\times25 = 1250 \)
For \( g(x) = 4^x \), substitute \( x = 5 \): \( g(5)=4^5 = 1024 \)

Step3: Calculate \( f(6) \) and \( g(6) \)

For \( f(x) = 50x^2 \), substitute \( x = 6 \): \( f(6)=50\times6^2 = 50\times36 = 1800 \)
For \( g(x) = 4^x \), substitute \( x = 6 \): \( g(6)=4^6 = 4096 \)

Step4: Calculate \( f(7) \) and \( g(7) \)

For \( f(x) = 50x^2 \), substitute \( x = 7 \): \( f(7)=50\times7^2 = 50\times49 = 2450 \)
For \( g(x) = 4^x \), substitute \( x = 7 \): \( g(7)=4^7 = 16384 \)

Step5: Analyze part (b)

Compare \( f(x) \) and \( g(x) \) for \( x\geq6 \). At \( x = 6 \), \( f(6)=1800 \), \( g(6)=4096 \); at \( x = 7 \), \( f(7)=2450 \), \( g(7)=16384 \). So \( f(x) \) is always less than \( g(x) \) for \( x\geq6 \).

Answer:

(for part a table):

\( x \)	\( f(x)=50x^2 \)	\( g(x)=4^x \)
4	800	256
5	1250	1024
6	1800	4096
7	2450	16384