To find the correct function representing the frog population after [tex]\( x \)[/tex] years given the initial population and the rate of decrease, we need to carefully analyze the situation.
1. Initial Population: The initial frog population is 1,200.
2. Decrease Rate: The population decreases at an average rate of 3% per year.
A decrease of 3% annually means that the population each year is 97% of the previous year's population. In mathematical terms, if [tex]\( P \)[/tex] is the population in one year, the population in the next year will be [tex]\( 0.97P \)[/tex].
### Step-by-Step Solution:
1. Starting Point:
- _Initial population_: [tex]\( P_0 = 1200 \)[/tex]
2. Population Decrease:
- Let's determine the population after 1 year:
[tex]\[
P_1 = 1200 \times 0.97
\][/tex]
- After 2 years:
[tex]\[
P_2 = 1200 \times 0.97 \times 0.97 = 1200 \times (0.97)^2
\][/tex]
- After 3 years:
[tex]\[
P_3 = 1200 \times (0.97)^3
\][/tex]
- Following this pattern, after [tex]\( x \)[/tex] years, the population will be:
[tex]\[
P_x = 1200 \times (0.97)^x
\][/tex]
3. Choose the Correct Function:
- The function that represents the frog population after [tex]\( x \)[/tex] years is:
[tex]\[
f(x) = 1200 \times (0.97)^x
\][/tex]
### Conclusion
Given the options:
- [tex]\( f(x) = 1,200(1.03)^x \)[/tex]
- [tex]\( f(x) = 1,200(0.03)^x \)[/tex]
- [tex]\( f(x) = 1,200(0.97)^x \)[/tex]
- [tex]\( f(x) = 1,200(0.97x) \)[/tex]
We can see that the correct function is:
[tex]\[ f(x) = 1,200 (0.97)^x \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{f(x) = 1,200(0.97)^x} \][/tex]
