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A coordinate plane has an increasing solid curve and a dotted straight line plotted on it, corresponding to two functions.
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The x-axis ranges from negative 4 to 52 in increments of 2 but the values are labeled in increments of 4. The y-axis ranges from negative 10 to 110 in increments of 10. The solid curve labeled Function number 1 passes through left parenthesis negative 2 comma .05 right parenthesis, left parenthesis 0 comma 1 right parenthesis, left parenthesis 4 comma 1.5 right parenthesis, left parenthesis 8 comma 2 right parenthesis, left parenthesis 20 comma 8 right parenthesis, left parenthesis 24 comma 10 right parenthesis, left parenthesis 32 comma 21 right parenthesis, left parenthesis 36 comma 30 right parenthesis, left parenthesis 40 comma 45 right parenthesis, and left parenthesis 49 comma 110 right parenthesis. The straight dotted line labeled Function number 2 passes through left parenthesis negative 2 comma negative 5, left parenthesis 0 comma 1 right parenthesis, left parenthesis 10 comma 21 right parenthesis, left parenthesis 24 comma 49 right parenthesis, left parenthesis 40 comma 81 right parenthesis, and left parenthesis 49 comma 99 right parenthesis. Both the curve and the line have arrows at both ends. The curve and the line intersect at left parenthesis 0 comma 1 right parenthesis and at left parenthesis 48 comma 97 right parenthesis. All coordinate values are approximate and the coordinates are unlabeled.


Given that the graph f(x)
is linear and the graph g(x)
is exponential solve the following problem:

Consider the graph of Function #1 and Function #2. How do the growth rates of the two functions compare?

(1 point)
Responses

Initially, Function #2 has the greater growth rate. After x=32
, the growth rate of Function #1 surpasses the growth rate of Function #2.
Initially, Function #2 has the greater growth rate. After x equals 32 , the growth rate of Function #1 surpasses the growth rate of Function #2.

The growth rate of Function #2 is always greater than the growth rate of Function #1.
The growth rate of Function #2 is always greater than the growth rate of Function #1.

Initially, Function #1 has the greater growth rate. After x=32
, the growth rate of Function #2 surpasses the growth rate of Function #1.
Initially, Function #1 has the greater growth rate. After x equals 32 , the growth rate of Function #2 surpasses the growth rate of Function #1.

The growth rate of Function #1 is always greater than the growth rate of Function #2.
The growth rate of Function #1 is always greater than the growth rate of Function #2.

All Answers 1

Answered by GPT-5 mini AI
Initially the linear function (Function #2) grows faster, but the exponential curve (Function #1) accelerates and overtakes it at about x = 32.

Reason: the line's slope ≈ (99 − (−5))/(49 − (−2)) ≈ 2.04, while the exponential's local rate around x = 32–36 is (30 − 21)/(36 − 32) = 2.25, so the exponential's growth rate surpasses the linear after about x = 32.