select the correct intervals of increase and decrease for the function $f(x)=\\log_{\\frac{1}{4}}(x - 4)+1$…

Explanation:

Step1: Analyze the base of the logarithm

The function is (y = \log_{\frac{1}{4}}(x - 4)+1). The base of the logarithm (a=\frac{1}{4}), and since (0 < a=\frac{1}{4}<1), the parent - function (y = \log_{a}u) is a decreasing function when (u>0).

Step2: Determine the domain of the function

For the function (y=\log_{\frac{1}{4}}(x - 4)+1), the argument of the logarithm (u=x - 4>0), so the domain is (x\in(4,\infty)).

Step3: Use the property of logarithmic functions

Let (u=x - 4). The function (y=\log_{\frac{1}{4}}u + 1). The derivative of (y) with respect to (x) (using the chain rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx})): (\frac{dy}{du}=\frac{1}{u\ln\frac{1}{4}}) and (\frac{du}{dx}=1). Since (\ln\frac{1}{4}<0) and (u=x - 4>0) for (x\in(4,\infty)), (\frac{dy}{dx}=\frac{1}{(x - 4)\ln\frac{1}{4}}<0) for all (x\in(4,\infty)).

Answer:

Decrease: ((4,\infty))