Let's detail the problem and solve it step by step.
We start with 12 fish in the pond at year 0. Each subsequent year, the number of fish doubles.
To find the number of fish after [tex]$x$[/tex] years, let's follow these steps:
### Step 1: Initial Analysis
- Year 0: There are 12 fish.
### Step 2: Progression Over the Years
- Year 1: The number of fish would be double that of year 0. So, [tex]\( 12 \times 2 = 24 \)[/tex] fish.
- Year 2: The number of fish would double again. From year 1 to year 2, [tex]\( 24 \times 2 = 48 \)[/tex] fish.
- Continuing in this way, we recognize that each year the number of fish multiplies by 2 compared to the previous year.
### Step 3: Generalizing for Year [tex]\( x \)[/tex]
We can see a pattern here. The number of fish after [tex]\( x \)[/tex] years is:
[tex]\[ 12 \times 2^x \][/tex]
### Step 4: Writing the Function
So, our function [tex]\( f(x) \)[/tex] to represent the number of fish after [tex]\( x \)[/tex] years is:
[tex]\[ f(x) = 12 \times 2^x \][/tex]
### Step 5: Matching with Given Options
We must now check the provided choices against our derived function.
A. [tex]\( f(x) = 12(2)^x \)[/tex]
B. [tex]\( f(x) = 12(x)^2 \)[/tex]
C. [tex]\( f(x) = 2(12)^x \)[/tex]
D. [tex]\( f(x) = 2(x)^{12} \)[/tex]
By inspection, choice A matches our derived function:
[tex]\[ f(x) = 12(2)^x \][/tex]
So, the function that represents the number of fish after [tex]\( x \)[/tex] years is:
[tex]\[ f(x) = 12(2)^x \][/tex]
Therefore, the correct answer is:
A. [tex]\( f(x) = 12(2)^x \)[/tex]
