Scientists released 6 rabbits into a new habitat in year 0. Each year, there were four times as many rabbits as the year before. How many rabbits were there after [tex]$x$[/tex] years? Write a function to represent this scenario.

A. [tex]$f(x)=6(4)^x$[/tex]
B. [tex][tex]$f(x)=4(x)^6$[/tex][/tex]
C. [tex]$f(x)=6(x)^4$[/tex]
D. [tex]$f(x)=4(6)^x$[/tex]



Answer :

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]