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

1 answer

To determine how the growth rates of the two functions compare, we need to understand the behavior of linear and exponential functions.

  1. Linear Function (Function #1): It grows at a constant rate. The slope of the line represents this constant growth rate.

  2. Exponential Function (Function #2): It grows at an increasing rate; its growth accelerates as x increases.

Initially, linear functions have a higher growth rate at smaller values of x, especially when the growth rate is moderate. However, exponential functions eventually surpass linear functions in growth due to their accelerating nature.

From the information provided:

  • Initially, Function #1 (linear) has the greater growth rate.
  • After a certain point (x=32 in this case), Function #2 (exponential) surpasses Function #1 in growth rate.

Thus, 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.