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Question

A coordinate plane has an increasing solid curve and a dotted straight line plotted on it, corresponding to two functions.

Given that the graph f(x)
is linear and the graph g(x)
is exponential solve the following problem:

Consider the graph of Function #1 and Function #2. How do the growth rates of the two functions compare?

(1 point)
Responses

Initially, Function #1 has the greater growth rate. After x=32
, the growth rate of Function #2 surpasses the growth rate of Function #1.
Initially, Function #1 has the greater growth rate. After x equals 32 , the growth rate of Function #2 surpasses the growth rate of Function #1.

The growth rate of Function #1 is always greater than the growth rate of Function #2.
The growth rate of Function #1 is always greater than the growth rate of Function #2.

The growth rate of Function #2 is always greater than the growth rate of Function #1.
The growth rate of Function #2 is always greater than the growth rate of Function #1.

Initially, Function #2 has the greater growth rate. After x=32
, the growth rate of Function #1 surpasses the growth rate of Function #2.

Answer 1

To analyze the growth rates of Function #1 (linear) and Function #2 (exponential), we can consider their characteristics:

1. Linear Function (Function #1): The growth rate remains constant; it increases by the same amount for each unit increase in \(x\).

2. Exponential Function (Function #2): The growth rate increases faster than linear growth; it grows by a percentage of its current value, causing it to surpass linear functions as \(x\) increases.

Given that the problem states that initially Function #1 has the greater growth rate, but eventually after a certain point \(x=32\), the growth rate of Function #2 surpasses the growth rate of Function #1, the correct response is:

Initially, Function #1 has the greater growth rate. After \(x=32\), the growth rate of Function #2 surpasses the growth rate of Function #1.