Use the table to answer the question.

Question

Term Number 	1	2	3	4	5	6
Sequence for
g(x)=20x
 	20	40	60	80	100	120
Sequence for
f(x)=2x
 	2	4	8	16	32	64
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 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 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 must have calculated the growth rates incorrectly. Both functions are growing at the same rate.
Zoya must have calculated the growth rates incorrectly. Both functions are growing at the same rate.

Zoya has it backward. The growth rate of f(x)=2x
 is always greater than the growth rate of g(x)=20x
.

GPT-4o mini · Answer

Zoya's mistake in comparing the growth rates of \( g(x) = 20x \) and \( f(x) = 2^x \) is that they are only comparing the values of each sequence at specific terms without considering the nature of exponential growth versus linear growth.

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 \( g(x) = 20x \).

This is because \( f(x) = 2^x \) grows exponentially, while \( g(x) = 20x \) grows linearly. In the long run, the exponential function will outgrow the linear function.