Answer
The domain of this function is $(-\infty,-1)\cup(5,\infty)$
Work Step by Step
$f(x)=\log\Big(\dfrac{x+1}{x-5}\Big)$
For this function to be defined, the expression inside the logarithm must always be positive and cannot be zero.
Solve the following inequality to obtain the domain of this function:
$\dfrac{x+1}{x-5}\gt0$
Set the numerator and the denominator equal to $0$ and solve each individual equation for $x$ to obtain the critical points:
$x+1=0$
$x=-1$
$x-5=0$
$x=5$
The critical points are $x=-1$ and $x=5$. These points divide the real line into the following intervals:
$(-\infty,-1)$, $(-1,5)$ and $(5,\infty)$
Elaborate a sign diagram (shown below) using test points within each of the intervals found and evaluating them in each factor.
It can be seen from the diagram that the inequality is satisfied in the intervals $(-\infty,-1)$ and $(5,\infty)$. The endpoints of these intervals do no satisfy the inequality and are to be left open.
The domain of this function is $(-\infty,-1)\cup(5,\infty)$
