To compare the growth rates of the two functions (a linear function represented by Function #1 and an exponential function represented by Function #2), we can analyze their behaviors.
-
Linear Function (Function #1): This function increases at a constant rate. For example, if the function has the form \( f(x) = mx + b \), where \( m \) is a constant slope, this means that for each unit increase in \( x \), \( f(x) \) increases by a constant amount \( m \).
-
Exponential Function (Function #2): This function increases at an increasing rate, meaning that as \( x \) increases, the rate of growth of the function also increases. An example of an exponential function could be \( g(x) = a \cdot b^x \), where \( b > 1 \). The rate of growth of \( g(x) \) will eventually outpace that of any linear function as \( x \) becomes sufficiently large.
Given this understanding, the typical observation is:
- Initially, the linear function may appear to have a higher growth rate compared to the exponential function, but as \( x \) reaches a certain point, the exponential function will surpass the linear function because it grows at an increasing rate.
From the options provided and the typical behavior of these functions, the most accurate response would be:
Initially, Function #1 has the greater growth rate. After \( x = 32 \), the growth rate of Function #2 surpasses the growth rate of Function #1.
1 answer