Question

A coordinate plane shows an increasing solid curve and a dotted straight line plotted on it, corresponding to two functions.
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 has it backward. The growth rate of f(x)=6⋅1.5x
f
(
x
)
=
6
⋅
1.5
x
is always greater than the growth rate of g(x)=5x+10
g
(
x
)
=
5
x
+
10
.
Marshall has it backward. The growth rate of f left parenthesis x right parenthesis equals 6 times 1.5 superscript x baseline is always greater than the growth rate of g left parenthesis x right parenthesis equals 5 x plus 10 .

The growth rate of g(x)=5x+10
g
(
x
)
=
5
x
+
10
is only greater than the growth rate of f(x)=6⋅1.5x
f
(
x
)
=
6
⋅
1.5
x
when x
x
is greater than approximately −1.3
−
1.3
.
The growth rate of g left parenthesis x right parenthesis equals 5 x plus 10 is only greater than the growth rate of f left parenthesis x right parenthesis equals 6 times 1.5 superscript x baseline when x is greater than approximately negative 1.3 .

Marshall is only considering a small portion of the domains of the functions. If he went farther to the left 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 negative values of x
x
.
Marshall is only considering a small portion of the domains of the functions. If he went farther to the left 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 rate of g left parenthesis x right parenthesis equals 5 x plus 10 for negative values of 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(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
.

Answer 1

Marshall's mistake lies in misunderstanding the growth rates of the two functions over different ranges. The correct response is:

**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 eventually surpasses the growth rate of g(x)=5x+10 for large values of x.**

The function \( f(x) = 6 \cdot 1.5^x \) is an exponential function, which grows faster than the linear function \( g(x) = 5x + 10 \) as \( x \) increases. While it may appear that \( g(x) \) is growing faster for some small values of \( x \), over time (as \( x \) approaches infinity), \( f(x) \) will eventually outpace \( g(x) \).