Answer: The required function is,
![f(x)=582(7^\frac{1}{6})^{6x}](https://tex.z-dn.net/?f=f%28x%29%3D582%287%5E%5Cfrac%7B1%7D%7B6%7D%29%5E%7B6x%7D)
Step-by-step explanation:
Since, the population which is increasing with the constant factor is,
![P(x)=ab^x](https://tex.z-dn.net/?f=P%28x%29%3Dab%5Ex)
Where, a is the initial population,
b is the growth factor per period,
x is the number of periods.
Here, the given function that represents the mosquito population after x years,
![f(x)=582(7)^x](https://tex.z-dn.net/?f=f%28x%29%3D582%287%29%5Ex)
Where, 7 is the growth factor per year,
Since, 12 months = 1 year,
2 month = 1/6 year.
Also, we can write,
![582(7)^x=582(7)^{\frac{1}{6}\times 6x}=582(7^\frac{1}{6})^{6x}](https://tex.z-dn.net/?f=582%287%29%5Ex%3D582%287%29%5E%7B%5Cfrac%7B1%7D%7B6%7D%5Ctimes%206x%7D%3D582%287%5E%5Cfrac%7B1%7D%7B6%7D%29%5E%7B6x%7D)
![\implies f(x)=582(7^\frac{1}{6})^{6x}](https://tex.z-dn.net/?f=%5Cimplies%20f%28x%29%3D582%287%5E%5Cfrac%7B1%7D%7B6%7D%29%5E%7B6x%7D)
Which is the required equivalent function that shows the population of mosquito is increasing in every 2 month by the growth factor
.