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

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.

Answer 1

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

As \(x\) increases, exponential functions grow significantly faster than linear functions. Though in the early terms (as shown in the table), \(g(x) = 20x\) appears to yield larger values, as \(x\) continues to increase, the terms of \(f(x) = 2^x\) will eventually surpass those of \(g(x) = 20x\).

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