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A coordinate plane shows an increasing solid curve and a dashed line plotted on it, corresponding to two functions.
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The x axis ranges from 0 to 3.2 in increments of 0.2 and the y axis ranges from 0 to 20 in increments of 2. The curve labeled f of x equals 1.8 to the power of x passes through left parenthesis 0 comma 1 right parenthesis, left parenthesis 1 comma 1.8 right parenthesis, left parenthesis 2 comma 3.24 right parenthesis, approximately left parenthesis 2.7 comma 4.8 right parenthesis, and left parenthesis 3 comma 5.832 right parenthesis. The line labeled g of x equals 1.8 times x passes through left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 1.8 right parenthesis, left parenthesis 2 comma 3.6 right parenthesis, approximately left parenthesis 2.7 comma 4.8 right parenthesis, and left parenthesis 3 comma 5.4 right parenthesis.


Image Long Description The x axis ranges from 0 to 3.2 in increments of 0.2 and the y axis ranges from 0 to 20 in increments of 2. The curve labeled f of x equals 1.8 to the power of x passes through left parenthesis 0 comma 1 right parenthesis, left parenthesis 1 comma 1.8 right parenthesis, left parenthesis 2 comma 3.24 right parenthesis, approximately left parenthesis 2.7 comma 4.8 right parenthesis, and left parenthesis 3 comma 5.832 right parenthesis. The line labeled g of x equals 1.8 times x passes through left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 1.8 right parenthesis, left parenthesis 2 comma 3.6 right parenthesis, approximately left parenthesis 2.7 comma 4.8 right parenthesis, and left parenthesis 3 comma 5.4 right parenthesis.

Austin is using graphs to compare the growth rates of g(x)=1.3x
and f(x)=1.3x
. Which statement is true?

(1 point)
Responses

The growth rate of f(x)=1.8x
is greater than the growth rate of g(x)=1.8x
when x is less than 1.
The growth rate of f of x is equal to 1 point 8 to the x th power is greater than the growth rate of g of x is equal to 1 point 8 x when x is less than 1.

The growth rate of f(x)=1.8x
is less than the growth rate of g(x)=1.8x
until x is approximately 2.7.
The growth rate of f of x is equal to 1 point 8 to the x th power is less than the growth rate of g of x is equal to 1 point 8 x until x is approximately 2.7.

The growth rate of f(x)=1.8x
is greater than the growth rate of g(x)=1.8x
when x is greater than approximately 2.2.
The growth rate of f of x is equal to 1 point 8 to the x th power is greater than the growth rate of g of x is equal to 1 point 8 x when x is greater than approximately 2.2.

The growth rate of g(x)=1.8x
is only greater than the growth rate of f(x)=1.8x
when x is between 1 and approximately 2.7.
The growth rate of g of x is equal to 1 point 8 x is only greater than the growth rate of f of x is equal to 1 point 8 to the x th power when x is between 1 and approximately 2.7.
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All Answers 1

Answered by GPT-5 mini AI
Answer: The third statement.

Reason: f'(x) = (1.8)^x ln(1.8) and g'(x)=1.8. At x=1, f'(1) ≈ 1.059 < 1.8; at x=3, f'(3) ≈ 3.43 > 1.8. The exponential's slope overtakes the linear slope between x=1 and x=3 (about x ≈ 1.9), so for x larger than about 2.2 the growth rate of f(x)=1.8^x is greater than that of g(x)=1.8x.