4. Marshall is comparing the growth rates of f(x) = 6(1.5)^x and g(x) = 5x + 10 using this graph. Based on the graph, he concludes that the growth rate of g(x) = 5x + 10 is always greater than the growth rate of f(x) = 6(1.5)^x. Where is his mistake?

Question

A) Marshall is backward. The growth rate of f(x) = 6(1.5)^x is always greater than the growth rate of g(x) = 5x + 10
B) Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f(x) = 6(1.5)^x eventually surpasses the growth rate of g(x) = 5x + 10 for negative values of x.
C) The growth rate of g(x) = 5x + 10 is only greater than the growth rate of f(x) = 6(1.5)^x when x is greater than approximatel -1.3
D) Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f(x) = 6(1.5)x eventually surpasses the growth rate of g(x) = 5x + 10 for large values of x.

Answer 1

To identify Marshall's mistake, let's analyze the growth rates of the two functions, \( f(x) = 6(1.5)^x \) and \( g(x) = 5x + 10 \).

1. Exponential Growth vs. Linear Growth: The function \( f(x) = 6(1.5)^x \) represents an exponential growth function, which generally outgrows any polynomial or linear function, such as \( g(x) = 5x + 10 \), as \( x \) increases.

2. Behavior of the Functions:

   • For large positive values of \( x \), \( f(x) \) will eventually surpass \( g(x) \) because exponential functions grow faster than linear functions as \( x \) becomes large.
   • For small values of \( x \) (including negative values), \( g(x) \) may be larger than \( f(x) \) initially, since exponential functions start from a smaller value close to 0 and grow from there.

Given these considerations, let's analyze the options provided:

A) Marshall is backward. The growth rate of \( f(x) = 6(1.5)^x \) is always greater than the growth rate of \( g(x) = 5x + 10 \).
This is incorrect because \( f(x) \) will eventually surpass \( g(x) \) for large \( x \), but it does not always do so for all \( x \).

B) Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of \( f(x) = 6(1.5)^x \) eventually surpasses the growth rate of \( g(x) = 5x + 10 \) for negative values of \( x \).
This option is incorrect because \( f(x) \) does not surpass \( g(x) \) for negative \( x \), but rather for large positive \( x \).

C) The growth rate of \( g(x) = 5x + 10 \) is only greater than the growth rate of \( f(x) = 6(1.5)^x \) when \( x \) is greater than approximately -1.3.
This statement may or may not be true, but it does not address the overall mistake of comparing growth rates incorrectly.

D) Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of \( f(x) = 6(1.5)^x \) eventually surpasses the growth rate of \( g(x) = 5x + 10 \) for large values of \( x \).
This is correct. It highlights that the large \( x \) regions aren't being considered, where \( f(x) \) does indeed eventually surpass \( g(x) \).

Thus, the correct answer is:
D) Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of \( f(x) = 6(1.5)^x \) eventually surpasses the growth rate of \( g(x) = 5x + 10 \) for large values of \( x \).