Let's work through the problem and derive the correct function step-by-step.
First, consider the initial number of termites, which is 12,000.
When the house is sprayed once, the number of termites is reduced to one-fourth of the initial number. Therefore, after one spray, the number of termites would be:
[tex]\[ \text{Number of termites after 1 spray} = 12,000 \times \frac{1}{4} \][/tex]
When the house is sprayed a second time, the number of termites is again reduced to one-fourth of the number from the previous spray. So after two sprays, the number of termites would be:
[tex]\[ \text{Number of termites after 2 sprays} = 12,000 \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) = 12,000 \times \left(\frac{1}{4}\right)^2 \][/tex]
Similarly, after three sprays, it would be reduced to:
[tex]\[ \text{Number of termites after 3 sprays} = 12,000 \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) = 12,000 \times \left(\frac{1}{4}\right)^3 \][/tex]
From this pattern, after [tex]\(x\)[/tex] sprays, the number of termites would be:
[tex]\[ \text{Number of termites after } x \text{ sprays} = 12,000 \times \left(\frac{1}{4}\right)^x \][/tex]
So the function [tex]\(f(x)\)[/tex] that represents the number of termites after [tex]\(x\)[/tex] sprays is:
[tex]\[ f(x) = 12,000 \left(\frac{1}{4}\right)^x \][/tex]
Comparing this to the given options, the correct answer is:
B. [tex]\(f(x) = 12,000\left(\frac{1}{4}\right)^x\)[/tex]
