Graph f(x) = log base 1/4 of x-1+1

Find where the expression ${\displaystyle {\log }_{\frac{1}{4}}(x-1)+1}$ is undefined.

Since ${\displaystyle {\log }_{\frac{1}{4}}(x-1)+1}$${\displaystyle \to }$${\displaystyle \infty }$ as ${\displaystyle x}$${\displaystyle \to }$${\displaystyle 1}$ from the left and ${\displaystyle {\log }_{\frac{1}{4}}(x-1)+1}$${\displaystyle \to }$${\displaystyle -\infty }$ as ${\displaystyle x}$${\displaystyle \to }$${\displaystyle 1}$ from the right, then ${\displaystyle x=1}$ is a vertical asymptote.

Ignoring the logarithm, consider the rational function ${\displaystyle R\left(x\right)=\frac{a{x}^n}{b{x}^m}}$ where ${\displaystyle n}$ is the degree of the numerator and ${\displaystyle m}$ is the degree of the denominator.

There are no horizontal asymptotes because ${\displaystyle Q\left(x\right)}$ is ${\displaystyle 1}$.

No oblique asymptotes are present for logarithmic and trigonometric functions.

This is the set of all asymptotes.

Replace the variable ${\displaystyle x}$ with ${\displaystyle 2}$ in the expression.

Simplify the result.

Convert ${\displaystyle 1}$ to decimal.

Replace the variable ${\displaystyle x}$ with ${\displaystyle 5}$ in the expression.

Simplify the result.

Convert ${\displaystyle 0}$ to decimal.

Replace the variable ${\displaystyle x}$ with ${\displaystyle 3}$ in the expression.

Simplify the result.

Convert ${\displaystyle \frac{1}{2}}$ to decimal.