To determine which function defines [tex]\((f \div g)(x)\)[/tex], let's start by analyzing the given functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ f(x) = (3.6)^{x+2} \][/tex]
[tex]\[ g(x) = (3.6)^{3x+1} \][/tex]
We need to find the function [tex]\((f \div g)(x)\)[/tex], which is the result of dividing [tex]\(f(x)\)[/tex] by [tex]\(g(x)\)[/tex]:
[tex]\[ (f \div g)(x) = \frac{f(x)}{g(x)} = \frac{(3.6)^{x+2}}{(3.6)^{3x+1}} \][/tex]
We can simplify this expression by using the properties of exponents. Specifically, the property [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[ (f \div g)(x) = (3.6)^{(x+2) - (3x+1)} \][/tex]
Next, simplify the exponent:
[tex]\[ (x+2) - (3x+1) = x + 2 - 3x - 1 = -2x + 1 \][/tex]
Therefore:
[tex]\[ (f \div g)(x) = (3.6)^{-2x + 1} \][/tex]
So, the function that defines [tex]\((f \div g)(x)\)[/tex] is [tex]\((3.6)^{-2x + 1}\)[/tex].
Comparing this result with the given options, we find that option B matches our result:
B. [tex]\((f \div g)(x) = (3.6)^{-2x + 1}\)[/tex]
Thus, the correct answer is:
B. [tex]$(f \div g)(x)=(3.6)^{-2 x+1}$[/tex]
