Answer: The required function is,
\(f(x)=582(7^\frac{1}{6})^{6x}\)
Step-by-step explanation:
Since, the population which is increasing with the constant factor is,
\(P(x)=ab^x\)
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,
\(f(x)=582(7)^x\)
Where, 7 is the growth factor per year,
Since, 12 months = 1 year,
2 month = 1/6 year.
Also, we can write,
\(582(7)^x=582(7)^{\frac{1}{6}\times 6x}=582(7^\frac{1}{6})^{6x}\)
\(\implies f(x)=582(7^\frac{1}{6})^{6x}\)
Which is the required equivalent function that shows the population of mosquito is increasing in every 2 month by the growth factor \(7^\frac{1}{6}\).
