Use the image to answer the question.

Question

A coordinate plane shows an increasing solid curve and a dotted straight line plotted on it.

The graph of f(x)=100⋅1.5x
 and g(x)=100x+100
 is shown. Which of the following statements is true?

Statement #1: While the growth rate of f(x)
 is initially greater than the growth rate of g(x)
, the growth rate of g(x)
 keeps increasing and eventually surpasses the growth rate of f(x)
.

Statement #2: The growth rate of g(x)
 is greater than the growth rate of f(x)
 between approximately x=0
 and x=4
.

Statement #3: While the growth rate of g(x)
 is initially greater than the growth rate of f(x)
, the growth rate of f(x)
 keeps increasing and, by x=2.5
, surpasses the growth rate of g(x)
.

(1 point)
Statement #
 is true.

GPT-4o mini · Answer

To determine which statement is true regarding the functions $f (x) = 100 \cdot {1.5}^{x}$ and $g (x) = 100 x + 100$ , we need to analyze their growth rates.

Statement #1: While the growth rate of $f (x)$ is initially greater than the growth rate of $g (x)$ , the growth rate of $g (x)$ keeps increasing and eventually surpasses the growth rate of $f (x)$ .
- This statement is incorrect because $f (x)$ is an exponential function which eventually grows faster than any linear function $g (x)$ as $x$ increases.
Statement #2: The growth rate of $g (x)$ is greater than the growth rate of $f (x)$ between approximately $x = 0$ and $x = 4$ .
- This statement is also incorrect. Generally, $g (x)$ has a constant growth rate of 100 (the slope of the line), whereas $f (x)$ grows at an increasing rate since it is an exponential function starting from $f (0) = 100$ and growing more rapidly as $x$ increases.
Statement #3: While the growth rate of $g (x)$ is initially greater than the growth rate of $f (x)$ , the growth rate of $f (x)$ keeps increasing and, by $x = 2.5$ , surpasses the growth rate of $g (x)$ .
- This statement is true. At small values of $x$ , $g (x)$ might appear to be increasing at a faster rate due to its linear nature, but because $f (x)$ is exponential, it will eventually outpace $g (x)$ as $x$ increases, and this point occurs somewhere after $x = 2.5$ .

Therefore, the correct choice is:

Statement #3 is true.