Question

select the correct intervals of increase and decrease for the function $f(x)=\log_{\frac{1}{4}}(x - 4)+1$
decrease: $(4,\infty)$
increase: $(-\infty,\infty)$
decrease: none
increase: none
increase: $(4,\infty)$

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: \((4,\infty)\)