Answer:
\(f(x)=2(1.17)^{7x}\) and growth rate is 17%.
Step-by-step explanation:
The general exponential growth function is
\(g(x)=a(1+r)^x\) ...... (1)
where, a is initial value, r is growth rate and x is time.
The given function is
\(f(x)=2(3)^{x}\)
Rose wants to manipulate the formula to an equivalent form that calculates seven times a year, not just once a year.
Using the properties of exponents the given function can be written as
\(f(x)=2(3^{\frac{1}{7}})^{7x}\)
\(f(x)=2(1.17)^{7x}\)
It can be rewritten as
\(f(x)=2(1+0.17)^{7x}\) .... (2)
On comparing (1) and (2) we get
r=0.17
Therefore, the required function is \(f(x)=2(1.17)^{7x}\) and growth rate is 17%.