Use the image to answer the question.

Question

Use the image to answer the question.

An increasing curve with an arrow on the upper end is plotted on a coordinate plane. The x-axis is labeled people and ranges from 0 to 5 in increments of 1. The y-axis is labeled money and ranges from 0 to 500 in increments of 100.

Mason is completing a study for his psychology course. For the study, he begins with $20 and asks individuals on the street if they want the money or if they would like him to double it and give it to the next person. He continues to do this until someone takes the money. This situation can be modeled by the exponential equation m(p)=20(2)p
, where m is the money and p is the number of people who opt to pass it on to the next person. Which of the following statements is true about the equation?

(1 point)
Responses

The domain (p-values) include the values greater than or equal to 20 (p≥20)
.
The domain ( p -values) include the values greater than or equal to 20 left parenthesis p greater than or equal to 20 right parenthesis .

As p increases, m decreases slowly at first and then quickly.
As p increases, m decreases slowly at first and then quickly.

The range (m-values) includes the values (m≥20)
.
The range ( m -values) includes the values left parenthesis m greater than or equal to 20 right parenthesis .

As p increases, m increases quickly at first and then slowly.
As p increases, m increases quickly at first and then slowly.

Answer 1

Bot GPT 4-o mini answered
2 hours ago

The equation given is $ m(p) = 20(2^p) $, where $ m $ represents the money as a function of $ p $, which is the number of people who opt to pass it on.

Let's break down the components of the equation:

Domain: The variable $ p $ (the number of people passing it on) can be any non-negative integer (0, 1, 2, ...), since $ p < 0 $ does not make sense in this context. Therefore, the domain of $ p $ is $ p \geq 0 $.
Range: The function depends on the exponential term $ 2^p $, which starts from $ 1 $ when $ p = 0 $ (resulting in $ m(0) = 20 $) and increases without bound as $ p $ increases. Thus, the minimum value of $ m $ is $ 20 $ (when $ p = 0 $) and there is no upper limit. This leads to the range $ m \geq 20 $.
Behavior as $ p $ increases: As $ p $ increases, $ m $ increases exponentially. Initially, when $ p $ is small, the increase might seem less significant, but it rapidly becomes significant as $ p $ increases due to the exponential nature of the function.

Given these insights, let’s evaluate the provided statements:

"The domain (p-values) include the values greater than or equal to 20 (p≥20)."
- False: The correct domain is $ p \geq 0 $.
"As p increases, m decreases slowly at first and then quickly."
- False: $ m $ is increasing, not decreasing.
"The range (m-values) includes the values (m≥20)."
- True: The range starts from $ 20 $ and goes to infinity.
"As p increases, m increases quickly at first and then slowly."
- False: $ m $ increases exponentially, not slowly.

Thus, the only true statement is: The range (m-values) includes the values (m≥20).