Question

The function f(x) = 20(3)^x represents the growth of a fish population every year in a local lake. Jesse wants to manipulate the formula to an equivalent form that calculates every half-year, not every year. Which function is correct for Jesse's purposes?

Answer 1

We need the exponents law

$a^{bx}=(a^b)^x$
So if 
a=3
b=1/2

then
$3^{1/2*x}=(3^{1/2})^x$
where x represent every half a year.

Answer 2

The correct answer is:

$f(x) = 20(2)^x$

Explanation:

This is an equation of the form

$f(x) = a\times b^x$,

where a is the initial population, x is the amount of time, and b = 1+r, where r is the rate at which the population increases per year.

Matching this with our function, we see that a = 20 and b = 3; this means 1+r=3,  so r = 2.  This means the rate at which the population increases is 200% per year.

This means that every half year, the population would increase by 200/2 = 100%; this means r = 1.  This gives us

$f(x) = a\times b^x\\\\f(x) = a\times (1+r)^x\\\\f(x)=20\times (1+1)^x\\\\f(x) = 20\times 2^x$