Given the information about the native bird population decreasing at a rate of 10% per year, let's define and solve it step-by-step.
Initially, you are given that the population after 1 year is 14,000 birds.
Therefore:
[tex]\[ f(1) = 14000 \][/tex]
For the recursively-defined function to describe how the bird population changes each subsequent year, you can use the fact that each year the population decreases by 10%. This means the population for any given year is 90% (or 0.90) of the population from the previous year.
Thus, the recursive function can be defined as:
[tex]\[ f(n) = f(n-1) \times 0.90 \][/tex]
for [tex]\( n \geq 2 \)[/tex].
Now, let's determine the bird population after 4 years.
- For Year 2:
[tex]\[ f(2) = f(1) \times 0.90 = 14000 \times 0.90 = 12600 \][/tex]
- For Year 3:
[tex]\[ f(3) = f(2) \times 0.90 = 12600 \times 0.90 = 11340 \][/tex]
- For Year 4:
[tex]\[ f(4) = f(3) \times 0.90 = 11340 \times 0.90 = 10206 \][/tex]
Thus, the number of birds remaining after 4 years is:
[tex]\[ 10206 \][/tex]
Summarizing the results:
[tex]\[ f(1) = 14000 \][/tex]
[tex]\[ f(n) = f(n-1) \times 0.90 \][/tex]
and after 4 years, 10206 birds will remain.
