To find the exponential function \( f(x) \) that represents the fox population decreasing by 8 percent per year, we start from the initial population and apply the decay formula.
The general form for an exponential decay function is:
\[ f(x) = P_0 \cdot (1 - r)^x \]
where:
- \( P_0 \) is the initial population,
- \( r \) is the decay rate (as a decimal),
- \( x \) is the number of years.
Given:
- \( P_0 = 367 \) (initial population),
- \( r = 0.08 \) (8 percent is converted to decimal).
The function becomes:
\[ f(x) = 367 \cdot (1 - 0.08)^x = 367 \cdot (0.92)^x \]
Now, to estimate the fox population 9 years from now, we will substitute \( x = 9 \) into the function:
\[ f(9) = 367 \cdot (0.92)^9 \]
First, we compute \( (0.92)^9 \):
\[ (0.92)^9 \approx 0.4224 \]
Now, multiply this value by the initial population:
\[ f(9) \approx 367 \cdot 0.4224 \approx 155.50 \]
Rounding to the nearest whole number, the estimated fox population 9 years from now is approximately \( 156 \).
Thus, the exponential function representing the fox population is:
\[ f(x) = 367 \cdot (0.92)^x \]
And the estimated fox population 9 years from now is:
\[ \approx 156 \]