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

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 #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.

Answer 1

To compare the growth rates of the two functions, we can analyze their characteristics based on the information provided.

1. Function #1 (linear): This function increases at a constant rate. The slope of the linear function represents its growth rate, which does not change regardless of the x-value.

2. Function #2 (exponential): An exponential function grows at an increasing rate, meaning that as x increases, the growth rate of this function also increases rapidly.

In general, exponential functions will have a greater growth rate than linear functions for sufficiently large x-values.

Given the statement "Initially, Function #2 has the greater growth rate. After x = 32, the growth rate of Function #1 surpasses the growth rate of Function #2," is incorrect since linear functions cannot surpass exponential functions in growth rate for large values of x.

Instead, the correct comparison should reflect that:

• Initially, Function #2 (exponential) has the greater growth rate.
• The growth rate of Function #2 (exponential) is always greater than that of Function #1 (linear) as x increases.

Therefore, the most accurate response is:

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