Zoya is comparing the growth rates of g(x)=20x

Question

Zoya is comparing the growth rates of  g(x)=20x
  and  f(x)=2x
  using their sequences as shown in the table. They notice that the terms of  f(x)=2x
  are always less than the terms of  g(x)=20x
  and so conclude that  f(x)=2x
  grows slower than  g(x)=20x
 . Where is their mistake in comparing the growth rates?

(1 point)
Responses

Zoya should have made a bigger table. If they did, they would have seen that the terms of f(x)=2x
 are eventually greater than the terms of g(x)=20x
.
Zoya should have made a bigger table. If they did, they would have seen that the terms of  f left parenthesis x right parenthesis equals 2 superscript x baseline  are eventually greater than the terms of  g left parenthesis x right parenthesis equals 20 x .

Zoya should be comparing the differences between the values of the terms in each sequence as opposed to just comparing the values of each sequence.
Zoya should be comparing the differences between the values of the terms in each sequence as opposed to just comparing the values of each sequence.

Zoya has it backward. The growth rate of f(x)=2x
 is always greater than the growth rate of g(x)=20x
.
Zoya has it backward. The growth rate of  f left parenthesis x right parenthesis equals 2 superscript x baseline  is always greater than the growth rate of  g left parenthesis x right parenthesis equals 20 x .

Zoya must have calculated the growth rates incorrectly. Both functions are growing at the same rate.

GPT-4o mini · Answer

Zoya's mistake in comparing the growth rates of \( g(x) = 20x \) and \( f(x) = 2^x \) lies in the nature of these functions. The function \( f(x) = 2^x \) is an exponential function, while \( g(x) = 20x \) is a linear function.

As \( x \) becomes large, the exponential function \( 2^x \) will eventually outgrow the linear function \( 20x \), regardless of the values observed in a limited table.

Therefore, the correct response is:

Zoya should have made a bigger table. If they did, they would have seen that the terms of \( f(x) = 2^x \) are eventually greater than the terms of \( g(x) = 20x \).