To determine the correct function that represents the number of rabbits after [tex]\(x\)[/tex] years, let's analyze the given scenario step-by-step.
1. Initial Condition:
At year 0, the number of rabbits released into the habitat was 6.
2. Growth Pattern:
Each year, the number of rabbits is quadrupled (i.e., multiplied by 4).
3. Function Identification:
Therefore, the population growth can be described using an exponential function. The general form of an exponential function is [tex]\(f(x) = a \cdot b^x\)[/tex] where:
- [tex]\(a\)[/tex] is the initial quantity (number of rabbits in year 0).
- [tex]\(b\)[/tex] is the growth factor (since the population quadruples each year, [tex]\(b = 4\)[/tex]).
- [tex]\(x\)[/tex] is the number of years.
Given that the initial quantity [tex]\(a = 6\)[/tex] and the growth factor [tex]\(b = 4\)[/tex]:
[tex]\[
f(x) = 6 \cdot 4^x
\][/tex]
Let's verify our identified function against the options provided:
A. [tex]\( f(x) = 6 \cdot 4^x \)[/tex]
B. [tex]\( f(x) = 4 \cdot x^6 \)[/tex]
C. [tex]\( f(x) = 6 \cdot x^4 \)[/tex]
D. [tex]\( f(x) = 4 \cdot 6^x \)[/tex]
Only option A matches the form [tex]\(f(x) = 6 \cdot 4^x\)[/tex]. Therefore, the correct function to represent the number of rabbits after [tex]\(x\)[/tex] years is:
[tex]\[
f(x) = 6 \cdot 4^x
\][/tex]
Hence, the correct answer is:
Option A. [tex]\(f(x) = 6 \cdot 4^x\)[/tex]
