Question

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

(1 point)
Responses

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⋅1.5x
𝑓
(
𝑥
)
=
6
⋅
1.5
𝑥
eventually surpasses the growth rate of g(x)=5x+10
𝑔
(
𝑥
)
=
5
𝑥
+
10
for large values of x
𝑥
.
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 left parenthesis x right parenthesis equals 6 times 1.5 superscript x baseline eventually surpasses the growth ra

Answer 1

answer

Answer 2

Marshall is correct to note that he has only considered a limited portion of the domains of the functions. The function \( f(x) = 6 \cdot 1.5^x \) is an exponential function, while \( g(x) = 5x + 10 \) is a linear function.

Exponential functions grow significantly faster than linear functions as \( x \) becomes large. Therefore, while \( g(x) \) may appear to have a greater growth rate for small values of \( x \), \( f(x) \) will eventually surpass \( g(x) \) for sufficiently large values of \( x \).

Thus, Marshall's mistake lies in not considering the long-term behavior of the functions; if he analyzed the graph further to the right, he would observe that the growth rate of \( f(x) \) surpasses that of \( g(x) \).

Answer 3

Marshall's mistake is that he 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 \( f(x) = 6 \cdot 1.5^x \) eventually surpasses the growth rate of \( g(x) = 5x + 10 \) for large values of \( x \).