Question

The function f(x) = 10(5)x represents the growth of a lizard population every year in a remote desert. Crista wants to manipulate the formula to an equivalent form that calculates every half-year, not every year. Which function is correct for Crista's purposes?

Answer 1

`f(x)=10\cdot (5^2)^{(x)/(2)}`

Explanation:

We have been given a function
`f(x)=10\cdot 5^x`, which represents represents the growth of a lizard population every year in a remote desert.

As Christa wants to write an an equivalent form that calculates every half-year, so we need to manipulate our function in such a way that when we substitute x = 1, the outer exponent becomes half.

That would be only possible, if we had x/2 in the outer exponent.

Using exponent property we can
`a^(b*c)=(a^b)^c` we can write a function that calculates the lizard population every half-year as:

`10\cdot (5)^x=10\cdot (5^2)^{(x)/(2)}`

Therefore, the function
`10\cdot (5^2)^{(x)/(2)}` represents the growth of lizard population every half year.

Answer 2

f(x) = 10(5)^x

t = 2x => x = t/2 => f(t) = 10(5)^(t/2)

Note that t = 1 => x =0.5 years; t = 2 => x = 1 year; t=3 => x = 1.5 years,...