A park ranger uses exponential functions to model the population of two species of butterflies in a state park.

The population of species A, x years from today, is modeled by function f.

f(x) = 1,400(0.70)x

The population of species B is modeled by function g, which has an initial value of 1,600 and increases by 20% per year.

Which statement correctly compares the functions modeling the two species?

A.

The populations of both species are increasing, but the population of species B is growing at a faster rate than species A.

B.

The population of species A is decreasing, and it had the greater initial population.

C.

The populations of both species are increasing, but the population of species A is growing at a faster rate than species B.

D.

The population of species A is decreasing, and it had the smaller initial population.

The statement that correctly compares the given functions is - 'The population of species A is decreasing, and it had the smaller initial population.'

The correct answer is an option (D)

What is an exponential function?

"A function of the form [tex]f(x)=b^x[/tex] where b is constant."

What is exponential growth formula?

" [tex]f(x) = a (1 + r)^x[/tex]

where a is the initial value

r is the growth rate

x is time"

For given question,

We have been given a exponential function [tex]f(x) = 1400(0.70)^x[/tex]

This function represents the population of species A, x years from today.

The population of species B is modeled by function g, which has an initial value of 1,600 and increases by 20% per year.

a = 1600

r = 20%

= 0.2

Using the exponential growth formula the exponential function that represents the population of species B would be,

[tex]g(x) = 1600 (1 + 0.2)^x\\\\g(x)=1600(1.02)^x[/tex]

We know that, if the factor b ([tex]f(x)=a\bold{b}^x[/tex]) is greater than 1 then the exponential function represents the growth and if b < 1 then the exponential function represents the decay of population.

From functions f(x) and g(x) we can observe that, the population of species A is decreasing, and it had the smaller initial population.

So, the correct answer is an option (D)

Learn more about the exponential function here:

https://brainly.com/question/11487261

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