To solve this problem, let’s break it down step-by-step and identify the function that models the bacterial population over time as well as its asymptotic behavior.
### Step 1: Initial Analysis
- We are given that the initial population of bacteria is 5,000.
- Each day, 40% of the bacteria remain alive, which implies that 60% of the bacteria die off each day.
### Step 2: Mathematical Representation
- The survival rate per day is 40%, which can be written as a decimal [tex]\(0.4\)[/tex].
- The number of bacteria remaining after [tex]\(x\)[/tex] days can be modeled using an exponential decay function.
### Step 3: Constructing the Function
- The initial amount of bacteria is [tex]\(5000\)[/tex].
- After [tex]\(1\)[/tex] day, the number of bacteria is [tex]\(5000 \times 0.4\)[/tex].
- After [tex]\(2\)[/tex] days, the number of bacteria is [tex]\(5000 \times (0.4)^2\)[/tex].
- After [tex]\(x\)[/tex] days, the number of bacteria is [tex]\(5000 \times (0.4)^x\)[/tex].
Thus, the function representing the number of live bacteria after [tex]\(x\)[/tex] days is:
[tex]\[ f(x) = 5000(0.4)^x \][/tex]
### Step 4: Analyzing the Asymptotes
- As [tex]\(x\)[/tex] increases, [tex]\( (0.4)^x \)[/tex] gets closer and closer to [tex]\(0\)[/tex] but never actually reaches it. This describes a horizontal asymptote.
- Therefore, the function has a horizontal asymptote at [tex]\( y = 0 \)[/tex].
### Step 5: Selection of the Correct Option
Now, let’s compare this with the given options:
1. [tex]\( f(x) = 5000(0.4)^x \)[/tex], with a horizontal asymptote of [tex]\( y = 0 \)[/tex]
2. [tex]\( f(x) = 5000(0.6)^x \)[/tex], with a vertical asymptote of [tex]\( x = 0 \)[/tex]
3. [tex]\( f(x) = 5000(1.4)^x \)[/tex], with a horizontal asymptote of [tex]\( y = 0 \)[/tex]
4. [tex]\( f(x) = 5000(1.6)^x \)[/tex], with a vertical asymptote of [tex]\( x = 0 \)[/tex]
Since we have determined that the function is [tex]\( f(x) = 5000(0.4)^x \)[/tex] with a horizontal asymptote at [tex]\( y = 0 \)[/tex], the correct option is:
[tex]\[ f(x) = 5000(0.4)^x, \text{ with a horizontal asymptote of } y = 0 \][/tex]
Thus, the best description for the graph of the function that represents the number of live bacteria after [tex]\(x\)[/tex] days of treatment is:
[tex]\[ f(x) = 5000(0.4)^x, \text{ with a horizontal asymptote of } y = 0 \][/tex]
