Question

given the functions $f(x) = 2x^5$ and $g(x) = 8 \cdot 4^x$, which of the following statements is true? answer $\circ$ $f(4) = g(4)$ $\circ$ $f(4) > g(4)$ $\circ$ $f(4) < g(4)$

Explanation:

Step1: Calculate \( f(4) \)

Substitute \( x = 4 \) into \( f(x)=2x^{5} \).
\( f(4)=2\times4^{5} \)
First, calculate \( 4^{5}=4\times4\times4\times4\times4 = 1024 \).
Then, \( f(4)=2\times1024 = 2048 \).

Step2: Calculate \( g(4) \)

Substitute \( x = 4 \) into \( g(x)=8\cdot4^{x} \).
\( g(4)=8\times4^{4} \)
Calculate \( 4^{4}=4\times4\times4\times4 = 256 \).
Then, \( g(4)=8\times256 = 2048 \). Wait, no, wait: \( 4^{4} \) is 256? Wait, no, \( 4^1 = 4 \), \( 4^2 = 16 \), \( 4^3 = 64 \), \( 4^4 = 256 \), \( 4^5 = 1024 \). Wait, no, when \( x = 4 \), \( g(x)=8\times4^{4} \)? Wait, no, the function is \( g(x)=8\cdot4^{x} \), so when \( x = 4 \), it's \( 8\times4^{4} \)? Wait, no, \( 4^4 = 256 \), so \( 8\times256 = 2048 \)? Wait, but \( f(4)=2\times4^{5}=2\times1024 = 2048 \)? Wait, that would mean \( f(4)=g(4) \)? But wait, let's recalculate:

Wait, \( f(x)=2x^5 \), so \( f(4)=2*(4)^5 \). \( 4^5 = 4*4*4*4*4 = 1024 \), so \( 2*1024 = 2048 \).

\( g(x)=84^x \), so \( g(4)=84^4 \). \( 4^4 = 256 \), so \( 8*256 = 2048 \). So \( f(4)=g(4) \). Wait, but maybe I made a mistake. Wait, no, \( 4^5 = 1024 \), \( 2*1024 = 2048 \). \( 4^4 = 256 \), \( 8*256 = 2048 \). So they are equal.

Wait, but let's check again. \( 4^5 = 44444 = (4^2)(4^2)4 = 16164 = 256*4 = 1024 \). Correct. \( 2*1024 = 2048 \).

\( 4^4 = 4*4*4*4 = 256 \). \( 8*256 = 2048 \). So yes, \( f(4)=g(4) \). Wait, but the options are \( f(4)=g(4) \), \( f(4)<g(4) \), \( f(4)>g(4) \). So the correct one is \( f(4)=g(4) \)? Wait, but let me check again. Wait, maybe I misread the functions. The function \( f(x)=2x^5 \), \( g(x)=8\cdot4^x \). So at \( x=4 \), \( f(4)=2(4)^5=21024=2048 \). \( g(4)=8(4)^4=8256=2048 \). So they are equal. So the correct statement is \( f(4)=g(4) \).

Answer:

\( f(4) = g(4) \) (the option with this statement)