Graph the function $f(x)=\log _{1 / 3}(x+1)$ by using transformations on the appropriate basic graph of the form $y=\log _{a} x$. State the domain and range of the function and the vertical asymptote of the graph.
Use the graphing tool to graph the equation.

Step 1 ：First, we need to understand the basic graph of the function \(y = \log_a x\). When \(a > 1\), the graph is increasing and when \(0 < a < 1\), the graph is decreasing. The graph always passes through the point \((1, 0)\) and has a vertical asymptote at \(x = 0\).

Step 2 ：Next, we look at the function \(f(x) = \log_{1/3}(x+1)\). This is a transformation of the basic graph where the \(x\) values are shifted left by 1 unit (due to the \(+1\) inside the logarithm) and the base of the logarithm is less than 1, so the graph is decreasing.

Step 3 ：Using a graphing tool, we can plot the function. The graph will start from the point \((-1, 0)\) and decrease as \(x\) increases. The vertical asymptote of the graph is at \(x = -1\).

Step 4 ：The domain of the function is the set of all \(x\) values for which the function is defined. Since the argument of the logarithm (\(x+1\)) must be greater than 0, the domain of the function is \(x > -1\), or \(x \in \boxed{(-1, \infty)}\) in interval notation.

Step 5 ：The range of a logarithmic function is always \(\boxed{(-\infty, \infty)}\), since a logarithm can take any real number as its value.